The Team aims at designing and developing constructive methods in modeling, identification and control of dynamical, resonant and diffusive systems.
Function theory and approximation theory in the complex domain, with applications to frequency identification of linear systems and 2D inverse boundary problems for the Laplace and Beltrami operators.
System and circuit theory with applications to the modeling of analog microwave devices. Development of dedicated software for the synthesis of such devices.
Inverse potential problems in 3D and harmonic analysis with applications to nondestructive control and magneto/electroencephalography.
Control and structure analysis of nonlinear systems with applications to orbit transfer of satellites.
Industrial collaboration under contract with Alcatel Alenia Space (Toulouse and Cannes), Thales AS (Paris), CNES (Toulouse), XLim (Limoges).
Exchanges with UST (Villeneuve d'Asq), University BordeauxI (Talence), University of Orléans (MAPMO), University of Pau (EPI INRIA Magic3D), University MarseilleI (CMI), CWI (the Netherlands), SISSA (Italy), the Universities of Illinois (UrbanaChampaign USA), California at San Diego and Santa Barbara (USA), Michigan at EastLansing (USA), Vanderbilt University (Nashville USA), Texas A&M (College Station USA), ISIB (CNR Padova, Italy), Beer Sheva (Israel), RMC (Kingston, Canada), University of Erlangen (Germany), Leeds (UK), Maastricht University (The Netherlands), Cork University (Ireland), Vrije Universiteit Brussel (Belgium), TUWien (Austria), TFHBerlin (Germany), Kingston (Canada), Szegëd (Hungary), CINVESTAV (Mexico), ENIT (Tunis), VUB (Belgium), KTH (Stockholm).
The project is involved in a EMS21RTG NSF program (with Vanderbilt University), in the INRIA Team Enée associated with LAMSINENIT (including the EPI Anubis and Poems), in an EPSRC Grant with Leeds University (UK), in the ERCIM “Working Group Control and Systems Theory”, in the ANR projects AHPI (Math., coordinator) and FILIPIX (Telecom.).
Modelingmeans abstracting the behavior of a phenomenon in terms of mathematical equations, and is generally used to predictthe outcome of this phenomenon. Modeling involves typically two steps: choosing a model class and estimating “optimal” parameters within this class. The model class mainly reflects the available knowledge of the underlying physical system, and the algorithmic effort the user is willing to pay. Indeed, the second step usually consists in approximating the experimental data by the prediction of the model. The ability to solve this approximation problem, which is often nontrivial and illposed, impinges on the effectiveness of a method.
Below, we shall be mainly interested in stable linear timeinvariant systems, in particular resonant ones, and also in isotropically diffusive systems.
If only finitely many real parameters need to be adjusted, as in the case of systems described by ordinary differential equations, this process is often called parametric identificationand it has long been the realm of the stochastic paradigm. When the parameters are sought in some functional space, the term inverse problemis most commonly used and the techniques dwell more on functional analysis. This clearcut distinction of course gets blurred in practice. In all cases numerical optimization has become an allpervasive tool.
The Apics team's research on this topic is mostly concerned with identifying elliptic differential equations from boundary data. Special emphasis is put on the functiontheoretic aspects of the approximation step. The basic features of our approach are perhaps best captured when discussing the classical example of harmonic identification for stable linear dynamical systems ( i.e.1D causal convolution operators), which is a standard issue in engineering practice. Here, one measures the response of the system to periodic excitations in its bandwidth, thereby estimating the Fourier transform of the kernel in some working range of frequencies. As the impulse response is supported on the positive timeaxis by causality, this Fourier transform extends analytically to the right halfplane, thereby defining the socalled transfer functionof the system. One then seeks a stable transferfunction, that is, a function analytic in the complex right halfplane, whose trace on the imaginary axis approximately meets that Fourier transform. No data are available outside the bandwidth, as high frequencies can seldom be measured and often show nonlinear behaviour. Thereby the identification problem translates into an inverse boundaryvalue problem for the equation, namely to recover a function, analytic in a plane domain, from (incomplete) boundary data. This point of view naturally extends to other elliptic partial differential equations in two dimensions, in particular to the Laplace and complex Beltrami equations. Motivated by free boundary problems in plasma control, inverse conductivity issues from impedance tomography, and questions of source recovery arising in magneto/electroencephalography, a good deal of the team's research on inverse problems today aims at generalizing this approach to the real Beltrami equation in dimension 2 and to the Laplace equation in dimension 3.
In many important applications of harmonic identification (in particular for resonant systems, see section ), one wants the model to be rational of suitable degree, either because this brings physical significance to the parameters which is necessary for design, or because the complexity must be kept reasonably low. Other structural constraints, arising from the physics of the phenomenon under study, often superimpose on the model: for instance a passive system must be modelled by a transferfunction of modulus at most 1, a passive network must have a symmetric scattering matrix, etc... This leads us to split this identification problem in two: first we seek a stable but infinite (numerically: high) dimensional model to fit the data, and second we approximate it by a lower order model reflecting further properties of the original system.
In some sense, the first step amounts to set up a reference behaviour for the model at eachfrequency. Mathematically speaking, it consists in reconstructing a function analytic in the right halfplane (the transfer function), from its values on an interval of the imaginary axis (the bandwidth). In other words, one wants to make the principle of analytic continuation effective from part of the boundary of the analyticity domain. This classical illposed issue ( i.e.the inverse Cauchy problem for the Laplace equation) we will embed into a family of wellposed extremal problems, that may be viewed as a regularization scheme of Tikhonovtype. These problems are infinitedimensional but convex. Note that, by the CauchyRiemann equations, the tangential derivative of the imaginary part of an analytic function is the normal derivative of its real part. The question above can therefore be recast as a DirichletNeumann problem with overdetermined data on part of the boundary. As such, it occurs in many more inverse problems than harmonic identification, and it arises naturally in higher dimensions when analytic functions get replaced by gradients of harmonic functions (see sections and ).
The second step aims at reducing the complexity while bringing physical significance to the design parameters. It typically consists of a rational approximation procedure with prescribed number of poles in certain classes of analytic functions. To make the best out of the parameters, it is usually of importance to compute truly optimal or at least nearoptimal approximants. Rational approximation in the complex domain is a classical but difficult nonconvex problem, for which few effective methods exist. In relation to system theory, two specific difficulties superimpose on the classical situation, namely one must control the region where the poles of the approximants lie in order to ensure the stability of the model, and one has to handle matrixvalued functions when the system has several inputs and outputs, in which case the number of poles must be replaced by the McMillan degree.
Below, rational or more generally meromorphic approximation with prescribed number of poles will be used to approach other inverse problems beyond harmonic identification. In fact, the way the singularities of the approximant ( i.e.its poles) relate to the singularities of the approximated function (that may include branchpoints, essential singularities etc...) is an allpervasive theme in approximation theory that receives much attention within the team. The bottom line is that, for appropriate classes of functions, the location of the poles of the approximant can be used as an estimator of the singularities of the approximated function (see sections and ). In this context, the analog of rational functions in higher dimensions consists of gradients of discrete Newtonian potentials.
We provide further details on the two steps mentioned above. in the subparagraphs to come, still stressing harmonic identification, in connection with the recovery of analytic functions, as an illustrative example.
Given a planar domain
D, the problem is to recover an analytic function from its values on a subset of the boundary of
D. In harmonic identification,
Dis the right halfplane and the analytic function is the transfer functions of a stable system. However, since the recovery issue may arise on domains with various shapes when it comes
to inverse problems, it is convenient to normalize
Dand apply in each particular case a conformal transformation to meet the normalization. In the simply connected case, which is that of the halfplane, we fix
Dto be the unit disk, so that its boundary is the unit circle
T. We denote by
H^{p}the Hardy space of exponent
pwhich is the closure of polynomials in the
L^{p}norm on the circle if
1
p<
and the space of bounded holomorphic functions in
Dif
p=
. Functions in
H^{p}have welldefined boundary values in
L^{p}(
T), which make it possible to speak of (traces of) analytic functions on the boundary.
A standard extremal problem on the disk is , , :
(
P_{0}) Let
1
p
and
fL^{p}(
T); find a function
gH^{p}such that
g
fis of minimal norm in
L^{p}(
T).
When seeking an analytic function in
Dwhich approximately matches some measured values
fon a subarc
Kof
T, the following generalization of (
P_{0}) naturally arises:
(
P) Let
1
p
,
Ka subarc of
T,
fL^{p}(
K),
and
M>0; find a function
gH^{p}such that
and
g
fis of minimal norm in
L^{p}(
K)under this constraint.
Here
is a reference behaviour capsulizing the expected behaviour of the model off
K, while
Mis the admissible error with respect to this expectation. The value of
preflects the type of stability which is sought and how much one wants to smoothen the data.
To fix terminology we generically refer to (
P) as a
bounded extremal problem. The solution to this convex infinitedimensional optimization problem can be obtained upon iteratively solving spectral equations for appropriate operators,
that involve a Lagrange parameter, with right handside given by the solution to (
P_{0}) for some weighted concatenation of
fand
,
. The underlying operator is of Toeplitz type when
p= 2, thereby stressing an unexpected link with Carleman's reconstruction formulas
. The operator is of Hankel type when
p=
, in which case the solution rests on the AdamjanArovKrein (in short: AAK)
theory. The latter connects the singular decomposition of Hankel operators to best meromorphic approximants in
, and would allow for us to treat the extended version of (
P) where
g, instead of being analytic, is allowed a prescribed number of poles in
D
. When
1<
p<
but
p2, the computations become more involved, as the duality mapping is nonlinear.
The situation when
p= 1is essentially open from the constructive viewpoint.
In all cases, the illposed character of analytic continuation is reflected by the denseness of traces of
H^{p}functions in
L^{p}(
K)(if
p=
this requires
fto be continuous on
K). It is to the effect that, if the data are not
exactlyanalytic, the constraint
becomes an equality and the approximation error
tends to 0 if, and only if,
Mgoes to infinity
. When
p= 2this phenomenon has been quantified according to the smoothness of
f, as estimates of the error as a function of
Mcan be derived from a diagonalization procedure for Toeplitz operators
. The error in fact decreases much faster, when
Mincreases, when the data are holomorphic in neighborhood of
D; this is of interest to discriminate between nearly analytic data and those lying far from representing a linear stable model.
Various modifications of
(
P)can be made to meet specific needs. For instance when dealing with lossless transfer functions (see section
), one may want to express the constraint on
in a pointwise manner and thus turn to
(
P^{'}) Let
1
p
,
Ka subarc arc of
T,
fL^{p}(
K),
and
; find a function
gH^{p}such that

g

Ma.e. on
and
g
fis of minimal norm in
L^{p}(
K)under this constraint.
Somewhat dual to (
P) is the extension problem where one seeks
such that the concatenated function
fhis as close as possible to
H^{p}under the constraint that
.
Yet another variation of (
P), aiming at solving inverse boundaryvalue problems from mixed DirichletNeumann data, goes as follows (see sections
and
):
(
P^{''}) Let
1<
p<
,
Kbe an arc of the unit circle
T,
fL^{p}(
K),
, and
M>0; find a function
gH^{p}such that
and
is of minimal norm in
L^{p}(
K)under this constraint.
When
p= 2, the solution to (
P^{''}) involves both Toeplitz and Hankel operators
). When
p= 1,
, the above formulation is illposed. When slightly modified, it gives rise to
a new identification technique of Robin coefficients, which is interesting in particular for corrosion control. see
.
For using
(
P),
(
P
^{'}), and (
P^{''}) to tackle inverse problems, it is natural to ask about the continuity of
gwith respect to
f,
and
M. As a general rule,
L^{p}continuity holds if
p<
, but only weak* continuity if
p=
Smoothness also is a delicate issue, for the solution is rather irregular at
the endpoints of
Kunless
Mand
are adjusted to
f. Sufficient conditions for smoothness form a topic of current research, in relation to the behaviour of Cauchy singular integrals on an arc at its endpoints. In the team's target
applications to harmonic identification of resonant systems, as implemented in PRESTOHF, the regular behaviour of such systems at high frequencies is used together with extension
formulations to circumvent this problem (see section
).
The abovementioned problems can be stated on an annular geometry rather than a disk. For
p= 2the solution proceeds much along the same lines
,
, although the spectral properties of the Toeplitz operators involved are rather different. When
Kis the outer boundary, (
P) regularizes a classical inverse problem occurring in nondestructive control, namely to recover a harmonic function on the inner boundary from overdetermined DirichletNeumann data on
the outer boundary (see section
). Interestingly perhaps, (
P^{''}) becomes a tool to approach Bernoulli type problems for the Laplacian, where overdetermined observations are made on the outer boundary and we
seek the inner boundaryknowing it is a level curve of the flux. In fact, solving (
P^{''}) with small
Mfor some initial guess of the inner boundary, and using the Lagrange parameter to locally deform this guess so as to improve the error, yields a descent algorithm tending either to a
solution or a topological singularity (see section
).
Considerable effort is currently payed by the team to carry over bounded extremal problems and their solution to more general settings. Such generalizations are twofold: on the one hand
APICS considers 2D diffusion equations with variable conductivity, on the other hand it investigates the ordinary Laplacian in
R^{n}. The applications we seek are to the determination of free boundaries in plasma control for the first case, and to source detection in electro/magnetoencephalography for the second
case.
A diffusion equation in dimension 2 can be viewed as the compatibility condition of a socalled real Beltrami equation
. This way analytic functions get replaced by quasiconformal maps in problems (
P_{0}) and (
P). Hardy spaces of solutions, which are a little more general than classical Sobolev solutions, have been introduced by the team when
1<
p<
to imbed such maps in
L^{p}(
T), and the illposedness of (
P) again translates into the denseness of solutions in
L^{p}(
K). The underlying Toepliz and Hankel operators, as well as the expansions of solutions needed to constructively handle such problems, are now under study. The goal
is to solve the analog of (
P^{''}) in this context to approach Bernoullitype problems (see section
).
At present, bounded extremal problems for the
nD Laplacian are considered on halfspaces or balls. Following
, Hardy spaces are defined as gradients of harmonic functions satisfying
L^{p}growth conditions on inner hyperplanes or spheres. Such a definition is especially suited for the study of DirichletNeumann problems. Toeplitz operators with scalar symbol have been
introduced to solve the analog of (
P) when
p= 2. From the constructive viewpoint, spherical harmonics offer a reasonable substitute to Fourier expansions
. Only very recently we defined operators of Hankel type that connect to the solution of (
P_{0}) in the BMO norm. The goal is to solve the analog of (
P) on spherical shells to approach inverse diffusion problems across a conductor layer.
Let as before
Ddesignate the unit disk,
Tthe unit circle, and
be the Hardy space of exponent
pof the
complementof
Din the complex plane. By definition,
h(
z)belongs to
if, and only if
h(1/
z)/
zbelongs to
H^{p}. We further put
R_{N}for the set of rational functions with at most
Npoles in
D, which allows us to define the meromorphic functions in
L^{p}(
T)as the traces of functions in
H^{p}+
R_{N}.
A natural generalization of problem (
P_{0}) is
(
P_{N}) Let
1
p
,
N0an integer, and
fL^{p}(
T); find a function
g_{N}H^{p}+
R_{N}such that
g_{N}
fis of minimal norm in
L^{p}(
T).
Problem (
P_{N}) aims at recovering a solution of the inhomogeneous Laplace equation
u=
_{N}, with
_{N}some (unknown) linear combination of
NDirac masses, from overdetermined DirichletNeumann data.
Only for
p=
and continuous
fis it known how to solve (
P_{N}) in closed form. The unique solution is given by the AAK theory, that allows one to express
g_{N}in terms of the singular vectors of the Hankel operator with symbol
f. The continuity of
g_{N}as a function of
fonly holds for stronger norms than uniform,
e.g.Besov norms
.
The case
p= 2is of special importance. Specifically, if we write
f=
f^{+}+
f^{}upon splitting the Fourier expansion of
fbetween coefficients of nonnegative and strictly negative index respectively, and if
r_{N}is a best approximant to
f^{}from
R_{N}in
L^{2}(
T), then
g_{N}=
f^{+}+
r_{N}. In particular when
, so that
f^{+}= 0, then (
P_{N}) reduces to rational approximation. Moreover, it turns out that
r_{N}has no pole outside
Dand vanishes at infinity, hence it is a
stablerational approximant to
f. However, in contrast with the situation when
p=
, this approximant may
notbe unique.
To perform the second step of the identification scheme described in section
, namely rational approximation with a prescribed number of poles to a function analytic in the right halfplane, one can
map the latter conformally to the complement of
Dand solve (
P_{N}) with
p= 2for the transformed function on
T. Although no demonstrably convergent algorithm is known, the former Miaou project (the predecessor of Apics) has designed a steepestdescent algorithm for the case
p= 2whose convergence to a
local minimumis guaranteed; it seems the only procedure meeting this property. Roughly speaking, it is a gradient algorithm that proceeds recursively with respect to the order
Nof the approximant, using the particular geometry of the problem to restrict the search to a compact region of the parameter space
. If there is no local
maximum, a property which is satisfied when
Nis large enough, then every local
minimumcan be obtained from
someinitial condition of lower degree. Still, it is not known whether the absolute
minimumcan always be obtained using the strategy of the Endymion or RARL2 software consisting in choosing initial conditions corresponding to
critical pointsof lower degree (see sections
and
). The algorithm, however, has proved rather effective in all applications carried out so far (see section
). It is interesting to note that when
fis rational (which is no real restriction in practice), finding all the critical points of (
P_{N}) when
p= 2is an algebraic problem that might, in principle, be handled
viaelimination theory. Despite the increase of computational power, though, such a procedure is far too complex for realistic values of
Nand of
deg
f(see section
).
The current approach is still not satisfactory in that the convergence to global minimum is not established in general. In this regard, APICS has undergone a longhaul study of the number
and nature of critical points, in which tools from differential topology and operator theory team up with classical approximation theory. The main discovery is that the nature of the critical
points (
e.g.
local minima, saddles...) depends on the decrease of the interpolation error to
fas
Nincreases. Besides, an index theorem can be proved
to the effect that, in the generic case where all critical points are nondegenerate, they are finite in number
and the sum of their indices add up to 1; recall that the index of a critical point is 1 if its Morse index is even and 1 if it is odd. Based on this, sufficient conditions have been
developed for a local
minimumto be unique. This technique requires strong error estimates that are often difficult to obtain, and most of the time only hold for
Nlarge. Examples where uniqueness or asymptotic uniqueness has been proved this way include transfer functions of relaxation systems (
i.e., Markov functions)
,
, the exponential function (the prototype of an entire function with convex Taylor coefficients), and meromorphic
functions (
à laMontessus de Ballore)
. The case where
fis the Cauchy integral on a hyperbolic geodesic arc of a Dinicontinuous function which does not vanish “too much” has been very recently answered in the positive, see section
. We conjecture that uniqueness is linked to the ratio of the tobeapproximated function and its derivative on the circle.
When the latter is greater than 1 (
i.e., the logarithmic variation is small), it has been proved that uniqueness indeed holds in degree 1
. The generalization to higher degree is an exciting open question.
An analog to AAK theory has been carried out for
2
p<
. Although not computationally as powerful, it has better continuity properties and stresses a continuous link
between rational approximation in
H^{2}(see section
) and meromorphic approximation in the uniform norm, allowing one to use, in either context, techniques available from the
other. Hence, similar to the case
p=
, the best meromorphic approximation with at most
npoles in the disk of a function
fL^{p}(
T)is obtained from the generalized singular vectors of a Hankel operator with symbol
f. This generalization has a strong topological flavor and relies on the critical points theory of LjusternikSchnirelman as well as on the particular geometry of the Blaschke products
of given degree.
A common feature to all these problems is that critical point equations express nonHermitian orthogonality relations for the denominator of the approximant. This is used in an essential manner to assess the behavior of the poles of the approximants to functions with branched singularities, which is of particular interest for inverse problems ( cf. sections , ).
When
1
p<2, problem (
P_{N}) is still fairly open.
In higher dimensions, the analog of problem (
P_{N}) is the approximation of a vector field with gradients of potentials generated by
Npoint masses instead of meromorphic functions. The issue is by no means understood at present, and is a major endeavour of future research problems.
Certain constrained rational approximation problems, of special interest in identification and design of passive system, arises when putting additional conditions on the approximant, for
instance that it should be smaller than 1 in modulus. How good the approximation to a given function
funder this constraint is unclear, since best approximants tend to violate the constraint by winding around
fon
T. The team begins to investigate such questions using hyperbolic interpolation. Specifically, in light of the progress made on classical Schur interpolation in the last years
, the convergence properties of the multipoint Schur interpolation are currently under study, see section
.
The introduction of a weight in rational approximation is another interesting issue, which is useful in order to balance the information with the noise. For instance in the stochastic theory, minimum variance identification leads to assign weights to the errors like the inverse of the spectral density of the noise. Extensions of the abovementioned algorithms were implemented in Endymion, based on the work in . The analysis of the critical points equations in the weighted case gives various counterexamples to unimodality in maximum likelihood identification .
Another kind of rational approximation problem arises in designproblems, that became over years an increasingly significant part of the team's activity (see sections , , and ). These are problems where constraints on the modulusof a rational function are sought, and they occur mainly in filter design where the response is a rational function of fixed degree (the complexity of the filter), analytic and bounded by 1 in modulus on the righthalfplane (passivity), whose modulus must be as close as possible to 1 on some subset of the imaginary axis (the passband) ans as close as possible to 0 on the complementary subset (the stopband).
When translated over to the circle, a prototypical formulation consists in approximating the modulus of a given function by the modulus of a rational function of degree
n, that is, to solve for
When
p= 2this problem can be reduced to a series of standard rational approximation problems, but usually one needs to solve it for
p=
. The case where

fis a piecewise constant function with values 0 and 1 can also be approached
viaclassical Zolotarev problems
, that can be solved more or less explicitly when the passband consists of a single arc. A constructive solution
in the case of several arcs (multiband filters) is one recent achievement of the team (see section
). Of course, though the modulus of the response is the first concern in filter design, the variation of the phase must
nevertheless remain under control to avoid unacceptable distortion of the signal. As a matter of fact, tradingoff abrupt changes in modulus for a smaller derivative of the phase is an
important issue in frequency design which currently under investigation within the team under contract with the CNES, see section
.
From the point of view of design, rational approximants are indeed useful only if they can be translated into physical parameter values for the device to be built. While such problems do
not pertain to rational approximation proper, they are of utmost importance in practice. Actually, the fact that a device's response is shaped in the frequency domain whereas the device
itself must be specified in the time domain is a major difficulty in the area that reflects the fundamental problem of harmonic analysis. This is where SystemTheory enters the scene, as the
correspondence between the frequency response (
i.e.the transferfunction) and the linear differential or difference equations that generate this response (
i.e.the statespace representation) is the object of the socalled
realizationprocess. Algebraically speaking, a realization of a rational matrix
Hof the variable
zis a 4tuple
(
A,
B,
C,
D)of real or complex matrices of appropriate sizes such that
H(
z) =
C(
zI
A)
^{1}B+
D.
Since filters have to be considered as multipoles, the issue must indeed be tackled in a matrixvalued context that adds to the complexity. A fair share of the team's research in this direction is concerned with finding realizations meeting certain constraints (imposed by the technology in use) for a transferfunction that was obtained with the abovedescribed techniques. The current approach is to solve algebraic equations in many variables using homotopy methods, which seems to be a pathbreaking methodology in the area of filter design (see section ).
We refer here to the behavior of the poles of best meromorphic approximants, in the
L^{p}sense on a closed curve, to functions defined as Cauchy integrals of complex measures whose support lies inside the curve. If one normalizes the contour to be the unit circle, one we
oneself again in the framework of sections
and
; the invariance of the problem under conformal mapping was established in
. 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 to obtain error rates as the degree goes large and also to tackle constructive issues like uniqueness. However, the original motivation of Apics is to consider this issue in connection with the approximation of the solution to a DirichletNeumann problem, so as to extract information on the singularities.
As a general rule, critical point equations for these problems express that the polynomial whose roots are the poles of the approximant is a nonHermitian orthogonal polynomial with respect to some complex measure (that depends on the polynomial itself and therefore varies with the degree) on the singular set of the function to be approximated. New results were obtained in recent years concerning the location of such zeroes. The approach to inverse problem for the Laplacian that we outline in this section appears to be attractive when the singularities are onedimensional, for instance in the case of a cracked domain (see section ). It can be used as computationally cheap preliminary step to obtain the initial guess of a heavier but more precise numerical optimization. It is rather complementary of the popular MUSICtype algorithms as it can in principle be used on a single stationary pair of DirichletNeumann data.
When the crack is sufficiently smooth, the approach in question is in fact equivalent to the 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 turns out to be the equilibrium distribution on the geodesic arc
of the Green potential and it charges the end points, that are
de factowell localized if one is able to compute sufficiently many zeros (this is where the method could fail). 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. We proved very recently that the poles tend to the contour
outside of which the function is analytic and singlevalued that minimizes the capacity of the condenser
, where
Tis the exterior boundary of the domain (paper in preparation, see section
). For the definition of a condenser and other basic facts from potential theory, see
.
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 (see section
). Concerning the problem of a general singularity, that may be two dimensional, one can formulate the following conjecture:
if
fis analytic outside and on the exterior boundary of a domain
Dand if
Kis the minimal compact set included in
Dthat minimizes the capacity of the condenser
(
T,
K)under the constraint that
fis analytic and singlevalued 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
npoles) of
fin
L^{p}(
T)
has its support in
Kand sweeps out on the boundary of
Kto the equilibrium distribution of the condenser
(
T,
K). This conjecture, which generalizes the abovementioned results on 1D singular sets, is far from being solved in general.
Results of this type open new perspectives in nondestructive control (see section ), in that they link issues of current interest in approximation theory (the behavior of zeroes of nonHermitian orthogonal polynomials) to some classical inverse problems for which a dual approach is thereby proposed: to approximate the boundary conditions and not the equation (as is classically done).
Let us point out that the problem of approximating, by a rational or meromorphic function, in the
L^{p}sense on the boundary of a domain, the Cauchy transform of a real measure, localized inside the domain, can be viewed as an optimal discretization problem for a logarithmic potential
according to a criterion involving a Sobolev norm. This formulation can be generalized to higher dimensions, even if the computational power of complex analysis is then no longer available,
and this makes for a longterm research project with a wide range of applications. It is interesting to mention that the case of sources in dimension three in a spherical geometry, can be
attacked with the above 2D techniques as applied to planar sections (see section
).
Matrixvalued approximation is necessary for handling systems with several inputs and outputs, and it generates substantial additional difficulties with respect to scalar approximation, theoretically as well as algorithmically. In the matrix case, the McMillan degree ( i.e., the degree of a minimal realization in the SystemTheoretic sense) generalizes the degree.
The problem we want to consider reads:
Let
and
nan integer; find a rational matrix of size
m×
lwithout poles in the unit disk and of McMillan degree at most
nwhich is nearest possible to
in
(
H
^{2})
^{m×
l}.Here the
L^{2}norm of a matrix is the square root of the sum of the squares of the norms of its entries.
The approximation algorithm designed in the scalar case generalizes to the matrixvalued situation
. The first difficulty consists here in the parametrization of transfer matrices of given McMillan degree
n, and the inner matrices (
i.e., matrixvalued functions that are analytic in the unit disk and unitary on the circle) of degree
nenter the picture in an essential manner: they play the role of the denominator in a fractional representation of transfer matrices using the socalled DouglasShapiroShields
factorization. The set of inner matrices of given degree has the structure of a smooth manifold that allows one to use differential tools as in the scalar case. In practice, one has to
produce an atlas of charts (parameterizations valid in a neighborhood of a point), and we must handle changes of charts 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 (see
sections
and
). The first one is integrated in the Endymion 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), XLim, and Alcatel Alenia
Space (Toulouse), on which they gave high quality results
in all cases encountered so far. These codes are now of daily use by Alcatel Alenia Space and XLim, coupled with
simulation software like EMXD to design physical coupling parameters for the synthesis of microwave filters made of resonant cavities (see section
).
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 (see section ).
Problems relative to multiple local minima naturally arise in the matrixvalued case as well, but deriving criteria that guarantee uniqueness is even more difficult than in the scalar case. The already investigated case of rational functions of the sought degree (the consistency problem) was solved using rather heavy machinery , and that of matrixvalued Markov functions, that are the first example beyond rational function has made progress only recently .
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
nby perturbation of a matrix of degree
n1where the drop in degree is due to a polezero 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.
In order to control a system, one generally relies on a model, obtained from a prioriknowledge, like physical laws, or from experimental observations. In many applications, it is enough to deal with a linear approximation around a nominal point or trajectory. However, there are important instances where linear control does not apply, either because the magnitude of the control is limited or because the linear approximation is not controllable. Moreover, certain control problems, such as path planning, are not local in nature and cannot be solved vialinear approximations.
Section describes a problem of this nature, where the controllability of the linear approximation is of little help. Besides, the structural study described in section aims at exhibiting invariants that can be used, either to bring the study back to that of simpler systems or to lay grounds for a nonlinear identification theory. The latter would give information on model classes to be used in case there is no a priorireliable information and still the blackbox linear identification is not satisfactory.
Stabilization by continuous state feedback (or output feedback which is a 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–loop system. One can consider this as a weak version of the optimal control problem which is to find a control that minimizes a given criterion (for instance the time to reach a prescribed state). Optimal control generally leads to a rather irregular dependence on the initial state; in contrast, stabilization is a qualitativeobjective ( i.e., to reach a given state asymptotically) which is more flexible and allows one to impose a lot more regularity.
Lyapunov functions are a wellknown tool to study the stability of noncontrolled dynamic systems. For a control system, a Control Lyapunov Functionis a Lyapunov function for the closedloop system where the feedback is chosen appropriately. It can be expressed by a differential inequality called the “Artstein (in)equation ”, that looks like the HamiltonJacobiBellmann equation but is largely underdetermined. 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 can 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 a joint collaborative research conducted with Alcatel Alenia Space (Cannes), where minimizing a certain cost is very important (fuel consumption / transfer time) while at the same time a feedback law is preferred because of robustness and ease of implementation (see section ).
Here we study certain transformations of models of control systems, or more accurately of equivalence classes modulo such transformations. The interest is twofold:
From the point of view of control, a command satisfying specific objectives on the transformed system can be used to control the original system including the transformation in the controller. 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, the interest is either to derive qualitative invariants to support the choice of a nonlinear model given the observations, or to contribute to a classification of nonlinear systems which is missing sorely today. Indeed, the success of the linear model in control and identification is due to the deep understanding one has of it. In the same manner, a more complete knowledge of invariants of nonlinear systems under basic transformations is a prerequisite for a more general theory of nonlinear identification.
Concerning the classes of transformations, a static feedbacktransformation of a dynamical control system is a (nonsingular) reparametrization of the control depending on the state, together with a change of coordinates in the state space. A dynamic feedbacktransformation of a control system consists of a dynamic extension (adding new states, and assigning them a new dynamics) followed by a state feedback on the augmented system. Let us now stress two specific problems that we are tackling.
Dynamic linearization.The problem of dynamic linearization, still unsolved, is that of finding explicit conditions on a system for the existence of a dynamic feedback that would make it linear. It is equivalent to the problem of finding a “parameterization” of the solutions by a certain number of functions that are not constrained by differential equations, i.e. a formula that gives the general solution as a function of these functions and their derivatives, i.e. a nonlinear differential operator whose order is sometimes called the order of the parameterization. Existence of such a parameterization has been emphasized over the last years as very important and useful in control. A system is differentially flatwhen such a parameterization exists, with the property that each solution in turn uniquely determines the parameterizing function, which is then obtained by applying a differential operator to the solution.
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. Thus, flatness turns out to be a property of the set of trajectories which is of importance for pathplanning and also gives a handle for tackling the problem of dynamic linearization.
An important question remains open: how can one algorithmically decide whether a given system has this property or not, i.e., is dynamically linearizable or not? This problem is both difficult and important for nonlinear control. The mathematical difficulty is that no a priori bound is known on the order of the above mentioned differential operator giving the parameterization. Within the team, results on low dimensional systems have been obtained , see also ; the above mentioned difficulty is not solved for these systems but results are given with artificially prescribed bounds on this order. These characterizations point at the complexity of the issue.
From the algebraicdifferential point of view, the module of differentials of a controllable system is free and finitely generated over the ring of differential polynomials in
d/
dtwith coefficients in the ring of functions on the system's trajectories. A basis for this module can be explicitly constructed
. The question is to find out if there exists a basis consisting of closed, 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 properties are required to ascertain that the degree of the solutions is finite, a midterm goal being to obtain a formal and implementable
algorithm to decide whether or not a given system is flat around a regular point. One can further consider interesting subproblems like deciding flatness with a given precompensator, or
characterizing “formal” flatness that would correspond to a weak interpretation of the differential equation. Such questions can also be raised locally, in the neighborhood of an equilibrium
point.
Topological Equivalence.In what precedes, we have not taken into account the degree of smoothnessof 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 locallyequivalent to a linear system in a neighborhood of an equilibrium (the HartmanGrobman theorem). It is thus tempting when classifying controlsystems to look for such equivalence modulo nondifferentiable transformations, hoping to bring about some robust “qualitative” invariants and perhaps stable normal forms. A HartmanGrobman theorem for control systems would say for instance, that outside a “meager” class of models (for instance, those whose linear approximation is noncontrollable), and locally around nominal values of the state and the control, no qualitative phenomenon can distinguish a nonlinear system from a linear one, all nonlinear phenomena being thus either of global nature or singularities. Such a statement is wrong: if a system is locally equivalent to a controllable linear system via a bicontinuous transformation—a local homeomorphism in the statecontrol space—it is alsoequivalent 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 noncausal transformations (see ) whose structure remains unclear, but acquires a concrete meaning when the entries are themselves generated by a finitedimensional 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 nonlinear system and that of its linear approximation when the latter is controllable?
Many systems coming from mathematical physics, applied mathematics and engineering sciences can be described by means of systems of ordinary or partial differential equations, difference equations, differential timedelay equations... In the case of linear systems, these systems can be defined by means of matrices with entries in noncommutative algebras of functional operators such as differential operators, shift operators, timedelay operators, difference operators...
The methods of
algebraic analysis
The research of APICS on such topics combines Gröbner bases techniques over some noncommutative polynomial rings with the development of new algorithms of algebraic
analysis in order to effectively check classical properties of module theory (e.g., existence of a nontrivial torsion submodule or
rpure torsion submodules, torsionfreeness, reflexiveness, projectiveness, stably freeness, freeness, simple or decomposable modules), give their systemtheoretical interpretations
(existence of autonomous elements or successive parametrizations, existence of minimal/injective parametrizations or Bézout equations) and compute important tools of homological algebra (e.g.,
(minimal) free resolutions, split long exact sequences, extension and torsion functors, projective and Krull dimensions, Hilbert power series). The developed algorithms are implemented in
various symbolic packages that are used to apply our results to systems theory (e.g., parameterizability, flatness, autonomous elements, equivalences of systems, factoring and decomposing
linear functional systems) and to mathematical physics (e.g., research of potentials, computations of the field equations and the conservation laws).
The bottom line of the team's activity is twofold, namely function theory and optimization in the frequency domain on the one hand, and the control of systems governed by differential equations on the other hand. Therefore one can distinguish between two main families of applications: one dealing with the design and identification of diffusive and resonant systems (these are inverse problems), and one dealing with the control of certain mechanical systems. For applications of the first type, approximation techniques as described in section allow one to deconvolve linear equations, analyticity being the result of either the use of Fourier transforms or the harmonic character of the equation itself. Applications of the second type mostly concern the control of systems that are “poorly” controllable, for instance low thrust satellites or optical regenerators. We describe all these below in more detail.
Localizing cracks, pointwise sources or occlusions in a 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 electro and magnetoencephalography as well as the detection of buried objects (mines, etc). However, no completely satisfactory 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. A technique that evolved from the singularvalue decomposition of the correlation matrix is popular in the field under the name of MUSICtype algorithms . The methods we describe are of a different nature, and they are especially valuable when no mutually independent timevarying measurements are available, either because the measurements are stationary or because only few measurements can be made, or else because the superposition of phenomena to be analyzed ( e.g.the superposition of several sources) are mutually correlated both in time and space. These methods can also be used to approach inverse free boundary problems of Bernoulli type (see section ).
The presence of cracks in a plane conductor, for instance, or of sources in a cortex (modulo a reduction from 3D to 2D, see below) can be expressed as a lack of analyticity of the (complexified) solution of the associated DirichletNeumann problem that may in principle be approached using techniques of best rational or meromorphic approximation on the boundary of the object (see sections , and ). In this connection, the realistic case where data are available on part of the boundary only offers 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
Kof the boundary
of a plane domain
D, the trace on
of a function
Fwhich is analytic in
Dexcept for a possible singularity across some subset
(typically, a crack or a discrete set of pointwise sources). One can then use the approximation techniques described above in order to:
extend
Fto all of
if the data are incomplete (it may happen that
Kwhen 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 in this way , often 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 analyzing. In particular, contacts with the Odyssée projectteam of Inria Sophia Antipolis has provided us with 3D magnetoencephalographic data from which 2D information was extracted (see section ). The team also made contact with other laboratories ( e.g., Vanderbilt University Physics Dept.) in order to work out 2D or 3D data from physical experiments.
The team is beginning to adapt such methods to problems with variable conductivity governed by a 2D Beltrami equation. The application we have in mind is to plasma confinement for thermonuclear fusion in a Tokamak, more precisely with the extrapolation of magnetic data on the boundary of the chamber from the outer boundary of the plasma, which is a level curve for the poloidal flux solving the original divgrad equation. Solving this inverse problem of Bernoulli type is of importance to determine the appropriate boundary conditions to be applied to the chamber in order to shape the plasma . A joint collaboration on this topic recently started with the Laboratoire J. Dieudonné at the University of Nice, and the CMILATP at the University of Marseille I. It is one of the collaborative research topics with S. Rigat as described in section . The goal is first to determine the shape of the surface of the plasma in the chamber from the outer boundary measurements, and in a second step to shape this boundary by choosing some appropriate magnetic flux on this outer boundary (see section ).
One of the best training grounds for the research of the team in function theory is the identification and design of physical systems for which the linearity assumption works well in the considered range of frequency, and whose specifications are made in the frequency domain. Resonant systems, either acoustic or electromagnetic based, are prototypical devices of common use in telecommunications. 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. Surface Acoustic Waves (in short: SAW) filters consist in a series of transducers (see figure ) 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 analytically the functions that can actually be realized for a given type of filter.
In the domain of space telecommunications (satellite transmissions), constraints specific to onboard technology lead to the use of filters with resonant cavities in the microwave range. These filters serve multiplexing purposes (before or after amplification), and consist of a sequence of cylindrical hollow bodies, magnetically coupled by irises (orthogonal double slits). The electromagnetic wave that traverses the cavities satisfies the Maxwell equations, forcing the tangent electrical field along the body of the cavity to be zero. A deeper study (of the Helmholtz equation) states that essentially only a discrete set of wave vectors is selected. In the considered range of frequency, the electrical field in each cavity can be seen as being decomposed along two orthogonal modes, perpendicular to the axis of the cavity (other modes are far off in the frequency domain, and their influence can be neglected).
Near the resonance frequency, a good approximation of the Maxwell equations is given by the solution of a second order differential equation. One obtains thus an electrical model for our filter as a sequence of electricallycoupled resonant circuits, and each circuit will be modeled by two resonators, one per mode, whose resonance frequency represents the frequency of a mode, and whose resistance represent the electric losses (current on the surface).
In this way, the filter can be seen as a quadripole, with two ports, when plugged on a resistor at one end and fed with some potential at the other end. We are then interested in the power
which is transmitted and reflected. This leads to defining a scattering matrix
S, that can be considered as the transfer function of a stable causal linear dynamical system, with two inputs and two outputs. Its diagonal terms
S_{1, 1},
S_{2, 2}correspond to reflections at each port, while
S_{1, 2},
S_{2, 1}correspond to transmission. These functions can be measured at certain frequencies (on the imaginary axis). The filter is rational of order 4 times the number of cavities (that is 16
in the example), and the key step consists in expressing the components of the equivalent electrical circuit as a function of the
S_{ij}(since there are no formulas expressing the lengths of the screws in terms of parameters of this electrical model). This representation is also useful to analyze the numerical
simulations of the Maxwell equations, and to check the design, particularly the absence of higher resonant modes.
In fact, resonance is not studied via the electrical model, but via a 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 for the transmission and the reflection (the modulus of he response being respectively close to 0 and 1 outside the interval measurement, cf.section ). This provides us with a scattering matrix of order roughly 1/4 of the number of data points.
Approximating this scattering matrix by a rational transferfunction of fixed degree (8 in this example) via the Endymion or RARL2 software ( cf.section ).
A realization of the transfer function is thus obtained, and some additional symmetry constraints are imposed.
Finally one builds a realization of the approximant and looks for a change of variables that eliminates nonphysical couplings. This is obtained by using algebraicsolvers and continuation algorithms on the group of orthogonal complex matrices (symmetry forces this type of transformation).
The final approximation is of high quality. This can be interpreted as a validation of the linearity hypothesis for the system: the relative
L^{2}error is less than
10
^{3}. This is illustrated by a reflection diagram (Figure
). Nonphysical couplings are less than
10
^{2}.
The above considerations are valid for a large class of filters. These developments have also been used for the design of nonsymmetric filters, useful for the synthesis of repeating devices.
The team investigates today the design of output multiplexors (OMUX) where several filters of the previous type are coupled on a common guide. In fact, it has undergone a rather general analysis of the question “How does an OMUX work?” With the help of numerical simulations and Schur analysis, general principles are being worked out to take into account:
the coupling between each channel and the “Tee” that connects it to the manifold,
the coupling between two consecutive channels.
The model is obtained upon chaining the corresponding scattering matrices, and mixes up rational elements and complex exponentials (because of the delays) hence constitutes an extension of the previous framework. Its study is being conducted under contract with Alcatel Alenia Space (Toulouse) (see sections and ).
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 similar ones continue to motivate research in control within the team. Generally speaking, aerospace engineering requires sophisticated control techniques for which optimization is often crucial, due to the extreme functioning conditions.
The team has been working for two years on control problems in orbital transfer with lowthrust engines, under contract with Alcatel Space Cannes (see section ). Technically, the reason for using these (ionic) low thrust engines, rather than chemical engines that deliver a much higher thrust, is that they require much less “fuel”; this is decisive because the total mass is limited by the capacity of the launchers: less fuel means more payload, while fuel represents today an impressive part of the total mass.
From the control point of view, the low thrust makes the transfer problem delicate. In principle of course, the control law leading to the right orbit in minimum time exists, but it is quite heavy to obtain numerically and the computation is nonrobust against many unmodelled phenomena. Considerable progress on the approximation of such a law by a feedback has been made, and numerical experiments have been conducted (see section ).
For more than 10 years, physicists have been working on the realization of elementary quantum gates with the goal to build in the future a quantum computer ( cf.the cavity quantum electrodynamics experiments with circular Rydberg atoms at the Ecole Normale Supérieure in Paris as well as the handling of trapped ions with lasers at Innsbruck University). The main difficulty to overcome for the effective construction of a quantum computer is the decoherence that results form the coupling of Qbits with their environment: entangled states are difficult to achieve and to maintain over a significant period of time. The goal is to adapt existing control techniques and if necessary to propose new ones for modeling and controlling open quantum systems. In particular, for Qbits coupled with the environment, controllability and disturbance rejection issues arise when trying to design a control that drives the system from one pure quantum state to another (quantum gate) while compensating for the decoherence induced by the environment.
In order to take decoherence into account, one has to use the Heisenberg point of view where the density matrix is used instead of the probability amplitude (Schrödinger point of view); this framework takes into account the coupling with a large environment (reservoir) and its irreversible effects. Under weak coupling and short environment autocorrelation time, the evolution can be described by a differential equation, called the master equationwhich has a welldefined structure under the socalled Lindblad operators ; it yields a finitedimensional bilinear control system, that has not been thoroughly studied up to now. This is the subject of ongoing research. A more sophisticated model is the BlochRedfield formalism ; it does not have a finitedimensional state (in the controltheoretic sense of this word), but it seems more realistic when the control undergoes fast variations. There is numerical evidence (see ) that, in this model, the control can effectively act against dissipation.
This is a very new research topic for the team. We report in section on some investigation that started within the postdoctoral stay of Hamza Jirari.
The development of the
RARL2 (Réalisation interne et Approximation Rationnelle L2) is a software for rational approximation (see section
)
http://
This software takes as input a stable transfer function of a discrete time system represented by
either its internal realization,
or its first
NFourier coefficients,
or discretized values on the circle.
It computes a local best approximant which is
stable, of prescribed McMillan degree, in the
L^{2}norm.
It is akin to the arl2 function of Endymion from which it differs mainly in the way systems are represented: a polynomial representation is used in Endymion, while RARL2 uses realizations, this being very interesting in certain cases. It is implemented in Matlab. This software handles multivariablesystems (with several inputs and several outputs), and uses a parameterization that has the following advantages
it incorporates the stability requirement in a builtin manner,
it allows the use of differential tools,
it is wellconditioned, and computationally cheap.
An iterative research strategy on the degree of the local minima, similar in principle to that of arl2, increases the chance of obtaining the absolute minimum (see section ) by generating, in a structured manner, several initial conditions.
RARL2 performs the rational approximation step in our applications to filter identification as well as sources or cracks recovery . It was released to the universities of Delft, Maastricht, Cork and Brussels. The paramatrization embodied in RARL2 was recently used for a multiobjective control synthesis problem provided by ESTECESA (The Netherlands) .
The identification of filters modeled by an electrical circuit that was developed by the team (see section
) has led to compute the electrical parameters of the underlying filter. This means finding a particular realization
(
A,
B,
C,
D)of the model given by the rational approximation step. This 4tuple must satisfy constraints that come from the geometry of the equivalent electrical network and
translate into some of the coefficients in
(
A,
B,
C,
D)being zero. Among the different geometries of coupling, there is one called “the arrow form”
which is of particular interest since it is unique for a given transfer function and also easily computed. The
computation of this realization is the first step of RGC. Subsequently, if the target realization is not in arrow form, one can nevertheless show that it can be deduced from the arrowform by a
complex orthogonal change of basis. In this case, RGC starts a local optimization procedure that reduces the distance between the arrow form and the target, using successive orthogonal
transformations. This optimization problem on the group of orthogonal matrices is nonconvex and has a lot of local and global minima. In fact, there is not always uniqueness of the realization
of the filter in the given geometry. Moreover, it is often interesting to know all the solutions of the problem, because the designer cannot be sure, in many cases, which one is being handled,
and also because the assumptions on the reciprocal influence of the resonant modes may not be equally well satisfied for all such solutions, hence some of them should be preferred for the
design. Today, apart from the particular case where the arrow form is the desired form (this happens frequently up to degree 6) the RGC software gives no guarantee to obtain a single
realization that satisfies the prescribed constraints. The software DedaleHF (see
), which is the successor of RGC, solves in a guaranteed manner this constraint realization problem.
PRESTOHF: a toolbox dedicated to lowpass parameter identification for microwave filters http://wwwsop.inria.fr/apics/personnel/Fabien.Seyfert/Presto_web_page/presto_pres.html. In order to allow the industrial transfer of our methods, a Matlabbased toolbox has been developed, dedicated to the problem of identification of lowpass microwave filter parameters. It allows one to run the following algorithmic steps, either individually or in a single shot:
determination of delay components, that are caused by the access devices (automatic reference plane adjustment);
automatic determination of an analytic completion, bounded in modulus for each channel,
rational approximation of fixed McMillan degree;
determination of a constrained realization.
For the matrixvalued rational approximation step, PrestoHF relies either on hyperion (Unix or Linux only) or RARL2 (platform independent), both rational approximation engines were developed within the team. Constrained realizations are computed by the RGC software. As a toolbox, PrestoHF has a modular structure, which allows one for example to include some building blocks in an already existing software.
The delay compensation algorithm is based on the following strong assumption: far off the passband, one can reasonably expect a good approximation of the rational components of
S_{11}and
S_{22}by the first few terms of their Taylor expansion at infinity, a small degree polynomial in
1/
s. Using this idea, a sequence of quadratic convex optimization problems are solved, in order to obtain appropriate compensations. In order to check the previous
assumption, one has to measure the filter on a larger band, typically three times the pass band.
This toolbox is currently used by Alcatel Space in Toulouse and a license agreement has been recently negotiated with Thales airborne systems. XLim (University of Limoges) is a heavy user of PrestoHF among the academic filtering community and some free license agreements are currently being considered with the microwave department of the University of Erlangen (Germany) and the Royal Military College (Kingstone, Canada).
We continued the development of Endymion, http://wwwsop.inria.fr/apics/endymion/index.html, a software licensed under the CeCILL license version two, see http://www.cecill.info. This software will offer most of the functionalities of hyperion (whose development has been abandoned in 2001). It is much more portable, not depending any more on an external garbage collector. Memory management has been made easier by adding a type PolynomL, that lies between Polynom (basic polynomial type) and Lisp (generic object); such objects are automatically destroyed after use. The rational approximation algorithm arl2and the bounded extremal problem bep2are implemented but not yet fully debugged.
DedaleHF is a software meant to solve exhaustively the coupling matrix synthesis problem in reasonable time for the users of the filtering community. For a given coupling topology the coupling matrix synthesis problem (C.M. problem for short) consists in finding all possible electromagnetic coupling values between resonators that yield a realization of a given filter characteristics (see section ). Solving the latter problem is crucial during the design step of a filter in order to derive its physical dimensions as well as during the tuning process where coupling values need to be extracted from frequency measurements (see Figure ).
DedaleHf consists in two parts: a database of coupling topologies as well as a dedicated predictorcorrector code. Roughly speaking each reference file of the database contains, for a given coupling topology, the complete solution to the C.M. problem associated to a particular filtering characteristics. The latter is then used as a starting point for a predictorcorrector integration method that computes the solution to the C.M. problem of the user, i.e.the one corresponding to a userspecified filter characteristics. The reference files are computed off line using Groebner basis techniques or numerical techniques based on the exploration a monodromy group. The use of such a continuation technique combined with an efficient implementation of the integrator produces a drastic reduction of the computational time, say, by a factor of 20.
Access to the database and integrator code is done via the web on http://wwwsop.inria.fr/apics/Dedale/WebPages. The software is free of charge for academical research purposes: a registration is however needed in order to have access to the tool's full functionality. Up to now 90 users have registered among the world (mainly: Europe, U.S.A, Canada and China) and 4000 reference files have been downloaded.
As mentioned in an extension of this software that handles symmetrical networks is under construction.
The OreModuleslibrary of Ore _{}algebra ( Ore _{}algebrais a part of the commercial release of Maple) is dedicated to the study of linear functional systems defined over certain Ore algebras of functional operators and their applications in mathematical systems theory and mathematical physics.
The main novelty of OreModulesis to combine the recent developments of the Gröbner bases over some noncommutative polynomial rings with new algorithms of algebraic analysis in order to effectively check classical properties of module theory (e.g., existence of a nontrivial torsion submodule, torsionfreeness, reflexiveness, projectiveness, stably freeness, freeness), give their systemtheoretical interpretations (existence of autonomous elements or successive parametrizations, existence of minimal/injective parametrizations or Bézout equations) and compute important tools of homological algebra (e.g., (minimal) free resolutions, split exact sequences, extension functors, projective or Krull dimensions, Hilbert power series).
The abstract language of homological algebra used in the algebraic analysis approach carries over to the implementations in OreModules: up to the choice of the domain of functional operators which occurs in a given system, all algorithms are stated and implemented in sufficient generality such that linear systems defined over the Ore algebras developed in the Maple package of Ore _{}algebraare covered at the same time. Applications of the OreModulespackage to mathematical systems theory are illustrated in a large library of examples.
The
Staffordpackage of
OreModulescontains an implementation of constructive versions of J. T. Stafford's famous but difficult theorem stating that every ideal over
the Weyl algebras
A_{n}(
k)and
B_{n}(
k)(
kis a field of characteristic 0) can be generated by two generators. Based on this implementation and on algorithmic results recently obtained by the authors of this package, two
algorithms have been implemented which compute bases of free modules over the Weyl algebras
and
.
The forthcoming QuillenSuslinpackage of OreModules, developed by A. Fabiańska (University of Aachen) with the help of A. Quadrat, contains an implementation of the famous QuillenSuslin theorem. In particular, this implementation allows us to compute bases of free modules over a commutative polynomial rings with coefficients in the field and in the principal ideal domain .
The Morphismspackage of OreModuleswas developed by T. Cluzeau (ENSIL, Limoges) and A. Quadrat in order to handle some homological tools such as computations of some morphisms between two finitely presented modules over Ore algebras, compute kernel, coimage, image and cokernel of such morphisms and projectors. Using the packages Staffordand QuillenSuslin, these results allow us to compute factorizations as well as finding some decompositions of linear systems over Ore algebras. In terms of module theory, the Morphismspackage gives some methods to test if two modules are isomorphic, if a given module contains a submodule (reducible modules) or if it can be written as the direct sum of two submodules. Applications of the Morphismspackage to mathematical physics and mathematical systems theory are illustrated in a library of examples.
The major use of
Tralicsremains the production of the RAWEB (Scientific Annex to the Annual Activity Report of Inria), as explained schematically on figure
. The input is a
A second application of
Tralicsis the production of metadata (author, title, keywords, abstracts) on HTML servers. One could imagine that a server like HAL computes all relevant meta data from the
The main philosophy of
Tralicsis to have the same parser as
Two major versions have been released this year, namely 2.10 in May (redesigned for the 2007 activity report) and 2.11 in November. The two reports
and
, were updated. Almost all math constructions found in the
Among the fine points that were enhanced this year, let us mention the following ones: the implementation of the
\perscommand, which was modified so that if a Team belongs to more than one Research Center, then you have to specify to which one each person belongs (a default value can be
indicated). The number of Hdr is counted. A major difference between previous versions is that no intermediate
The input encoding mechanism was also enhanced: it is possible to specify the encoding (via the inputenc package) anywhere in the document. The MathML processor has been enhanced, as described in . New packages, like braket have been implemented.
Solving overdetermined Cauchy problems for the Laplace equation on an annulus (in 2D) or a spherical layer (in 3D) in order to treat incomplete experimental data is a necessary ingredient of the team's approach to inverse source problems, in particular for applications to EEG since the latter involves propagating the initial conditions from the boundary to the center of the domain where the singularities ( i.e.the sources) are sought. Here, the domain is typically made of several homogeneous layers of different conductivities.
Solving Cauchy problems on a 2D annulus is the main topic of M. Mahjoub's PhD thesis. This issue arises when identifying a crack in a tube or a Robin coefficient on the inner skull thereof. It can be formulated as a best approximation problem on part of the boundary of a doubly connected domain, and both numerical algorithms and stability results were obtained in this framework , . They generalize those previously obtained in simply connected situations and have been efficiently coupled with the sources recovery issue in a 2D model of the EEG problem .
Still in the 2D case with incomplete data, the geometric problem of finding, in a stable and constructive manner, some unknown (insulating) part of the boundary of a domain was 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, here also provide us with interesting algorithms for the inverse problem under consideration.
Cauchy problems on 3D spherical layers offer an opportunity to state and solve extremal problems for harmonic fields for which an analog of the Toeplitz operator approach to bounded
extremal problems
has been obtained. More specifically, the density of traces of harmonic gradients in
L^{2}of an open subset of the 3D sphere was established, and a Toeplitz operator with symbol the characteristic function of such a subset was defined. Then, a best approximation on the
subset of a general vector field by a harmonic gradient under a
L^{2}norm constraint on the complementary subset can be computed by an inverse spectral equation for the abovementioned Toeplitz operator. Constructive and numerical aspects of the
procedure (harmonic 3D projection, Kelvin and Riesz transformation, spherical harmonics) are under study and encouraging results have been obtained on numerically simulated data
.
The corresponding problem in
is considerably more difficult. Very recently, in a collaborative work with A. Bonami and S. Grellier (université d'Orléans) in the framework of the ANR project AHPI, we obtained
that the BMO distance between a bounded vector field on the sphere and a bounded harmonic gradient is within a constant of the norm of a Hankellike operator, acting on
L^{2}divergencefree vector fields with values in
L^{2}gradients. Such Hankel operators were apparently not introduced before. This result, which is based on the atomic decomposition for
H^{1}and the divcurl lemma, opens new perspectives on the analogy between extremal problems on the disk and the sphere.
In collaboration with the CMILATP (University Marseille I) and in the framework of the ANR AHPI, the team considers 2D diffusion processes with variable conductivity. In particular the
socalled 2D
real Beltrami equation, whose compatibility condition is the 2D divgrad equation, was investigated. In the case of a smooth domain, and for a smooth (
) conductivity, we analyzed the Dirichlet problem for solutions in Sobolev
W^{1,
p}classes,
1<
p<
. The traces of such solutions on a strict subsets
Kof the boundary turn out to be dense in
W^{11/
p,
p}(
K). The behavior of the associated Cauchy and Beurling operators as well as
W^{1,
p}estimates for generalized Riesz transforms have already been obtained
,
. We also introduced less regular solutions of Hardytype (
i.e.having bounded integral
L^{p}means on a system of curves tending to the boundary). Their traces merely lie in
L^{p}of the boundary, a space which is better suited to identification from pointwise measurements than the more regular
W^{11/
p,
p}. Again these traces turn out to be dense on strict subsets of the boundary. This should allow for us to state Cauchy problems as bounded extremal issues in
L^{p}classes of generalized analytic functions, in a manner similar to what we did for analytic functions as discussed in section
. The case of a conductivity that is merely in
, which is important for inverse conductivity problems, is now under study. Then, it is still unknown whether solutions exist for all
p.
The application that initially motivated this work is described in the next section.
Let us briefly describe a potential application of inverse boundary problems for the Beltrami equation, on a 2D doubly connected domain, to plasma confinement for thermonuclear fusion in a Tokamak; this collaborative work was started in collaboration with the Laboratoire J. Dieudonné (University of Nice). In the particular case at hand, it seems possible to explicitly compute a basis of solutions (Bessel functions) that should greatly help the computations, see , but the techniques should be valuable more generally.
In the most recent Tokamaks, like Jet or ITER, an interesting feature of the level curves of the poloidal flux is the occurrence of a cusp (a saddle point of the poloidal flux, called an X point), and it is desirable to shape the plasma according to a level line passing through this X point for physical reasons relating to the efficiency of the energy transfer.
The problem we have in mind here is of dual Bernoulli type. Classically, the interior Bernoulli problem on a domain
(see
) is to find a closed subset
and a harmonic function
uin
such that
u= 0on
,
u= 1on
A, and
u/
=
Qon
A, where
Qis a given positive constant and
indicates the outer normal. A natural generalization is obtained on letting
usatisfy a more general diffusion equation
u) = 0
in , for some nonconstant conductivity >0.
The dual problem arises when both
uand
u/
nare given on the known boundary
while
u=
Qis constant on
A. Note that this issue is
overdetermined, that is, the boundary data on
have to satisfy some compatibility conditions (of generalized CauchyRiemann type). One motivation for the dual problem is the observation that, in the transversal section of a Tokamak
(which is a disk if the vessel is idealized into a torus), the socalled poloidal flux is subject to (
) outside the plasma volume for some simple explicit real analytic function
, while the boundary of the plasma is a level line of this flux
. Actually, when looking for a X point, the main interest is attached to the smallest connected socalled
“elliptic” solution, which makes for a definite object of study among all other solutions.
When
is constant and
u/
nhas zero mean on
, it is wellknown that
uhas a conjugate function
vsuch that
u+
ivis holomorphic in
. More generally, as soon as
is bounded away from zero and
u/
nhas zero mean on
, a generalized conjugate exists such that
f=
u+
ivsatisfies the socalled
real Beltrami equation:
where
= (1
)/(1 +
). Moreover, the DirichletNeumann data for
udetermine the boundary values of
fon
(up to an additive imaginary constant). For fixed
Aand
Q, we intend to study the extremal problem of best approximating these values by (the trace on
of) a solution to (
) under the constraint that it has nonnegative real part at most
Qon
A. This is an infinitedimensional convex problem whose Lagrange parameter will indicate both how to deform and how to modify
Alocally in order to improve the criterion.
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 the 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 section , allows one to efficiently locate their position. This is the topic of the article , where the related situation of small inhomogeneities connected to mine detection is also considered, and of for a 2D version of the EEG problem, where a first analytic completion step has been performed on the data.
The problem of sources recovery can be handled in 3D balls by using best rational approximation on 2D cross sections (disks) from traces of the boundary data on the corresponding circles. It turns out that each of these traces coincides with a 2D analytic functions in the slicing plane, that has branched singularities inside the disk . These singularities are related to the actual location of the sources (namely, they reach in turn a maximum in modulus when the plane contains one of the sources). Hence, we are back to the 2D framework where approximately recovering these singularities can be performed using best rational approximation.
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 crosssections are quadrature domains or Rdomains. In this framework, best rational approximation can still be performed in order to recover the singularities of solutions to Laplace equations, but complexity issues have to be examined carefully. The preliminary case of an ellipsoid requires the preliminary computation of an explicit basis of ellipsoidal harmonics , and is the topic of the PhD thesis of M. Zghal.
In 3D, epileptic regions in the cortex are often represented by pointwise sources that have to be localized from measurements on the scalp of a potential satisfying a Laplace equation (EEG, electoencephalography). Note that the patient's head is here modeled as a nested sequence of spherical layers. A breakthrough was made which makes it possible now to proceed via best rational approximation on a sequence of 2D disks along the inner sphere . 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 which makes for an easy detection; when there are several sources, their localisation requires a slightly more sophisticated machinery to make the convergence of poles of meromorphic approximants effective (see section ). This and other related issues including some preprocessing of the function are still under study. This inverse EEG problem is the object of a collaboration between the Apics and Odyssée Teams, through the ACI “ObsCerv” during last years and now around the associate engineer R. Bassila. Her task consists in achieving a dedicated numerical interface between available softwares from both teams, “FindSources3D”.
We mentioned in section the role of parameters defining an atlas of charts in optimization problems on a manifold. Such parametrizations are available for lossless transfer functions and have been implemented in our rational approximation softwares. Charts were obtained from Schur analysis, the parameters being NevanlinnaPick interpolation values . These parametrizations present a number of advantages from an algorithmic viewpoint : flexibility, implementation facilities, numerical behavior. Our results concerning more general interpolation parameters, associated with a Nudelman (contour integral) interpolation problem have been published this year . This last parametrization has been used in an application to multiobjective control which is the object of a contract (n ^{o}20084/06/NL/JD) between CMAEMP and ESTCESA (Noordwijk, NL). The problem of multiobjective control synthesis leads to a linear matrix inequality which has to be minimized with respect to a fixed degree stable system called the Youla parameter. The parameterization is used to control the poles part of the Youla parameter. The linear part is obtained by the semidefinite minimization of the LMI.
Now, the balanced realizations obtained in an arbitrary chart possess no particular structure. However, a particular form of the realization matrix can present some interest either because it explicitly contains the physical parameters (coupling parameters between resonators for the coupling matrix of an hyperfrequency filters, see section ) or because of its nice behaviour under truncation. We have some results in that connection. Upon choosing the interpolation points at zero and the directions in a particular manner among standard basis vectors, a minimal atlas can be specified in which the balanced realizations have a staircase structure with the property that the corresponding controllability matrix is positive uppertriangular .
Up to now, our parametrizations only concern discretetime transfer functions. In continuoustime, which is relevant in many applications, we have established a connection in the SISO case between the wellknown Schwartz form and a boundary interpolation problem on the imaginary axis. These results have been presented at the Benelux meeting on systems and control (Lommel, Belgium, March 2007).
Darlington Synthesis is an old problem dating back to pretransistor times (1939), and it was studied in a system theoretic framework in the seventies. In recent years, with the advent of the information technology society, synthesis problems became important again. Indeed, new devices are being built whose design rests on optimizing a linear finitedimensional approximation around the nominal frequency. These models are subject to various constraints coming from the underlying physical device, and parametrizing them is a prerequisite to optimization, see section for the example of resonant cavities.
Darlington Synthesis turns out to show up naturally when modelling SAW filters, which are used in most cheap telecommunication devices (e.g. mobile phones)
. In this application the physical law of reciprocity induces a symmetry constraint. In mathematical terms, given a
p×
psymmetric Schur (contractive) function
S, the problem is to imbed
Sinto symmetric lossless function
of twice the original size. One of course looks for a minimal degree completion, or dually one imposes the degree to be minimal and seeks a minimal size completion. In collaboration with
P. Enqvist from KTH (Stockholm, Sweden), we characterized the existence of a
(2
p×2
p)symmetric Darlington synthesis with specified increase of the McMillan degree: a symmetric completion of a symmetric contractive matrix
Sof degree
nexists in degree
n+
kif, and only if,
I
SS^{*}has at most
kzeros with odd multiplicity
. In the language of circuit theory, this results tells us about the minimal number of gyrators to be used in
circuit synthesis. Using a frequency domain approach, these results have been extended to the case of realvalued functions. These results have been presented at LynSys2007 (Canberra,
February). In view of multiport synthesis applications, the completion of a scalar Schur function, or of a unitary row, to a square conservative matrix is currently under study.
The results of
and
have been extensively used over the last two years to prove the convergence in capacity of
L^{p}best meromorphic approximants on the circle (
i.e.solutions to problem (
P_{N}) of section
) when
p2, for those functions
fthat can be written as Cauchy transforms of complex measures supported on a hyperbolic geodesic arc
. A rational function can also be added to
fwithout modifying the results, which is useful for applications to inverse sources problems. Some mild conditions (bounded variation of the argument and powerthickness of the total
variation) were required on the measure. Here, we recall that convergence in capacity means that the (logarithmic) capacity of the set where the error is greater than
goes to 0 for each fixed
>0. Deepening the analysis, we were able this year to quantify this
convergence, namely it is geometric with pointwise rate
exp{1/
C
G}where
Cis the capacity of the condensor
and
Gthe Green potential of the equilibrium measure. The results can be adapted to somewhat general interpolation schemes, and two articles were submitted for publication on the subject
,
. From this work it follows that the counting measures of the poles of the approximants converge, in the weak*
sense, to the Green equilibrium distribution on
. In particular the poles cluster to the endpoints of the arc, which is of fundamental use in the team's approach to source detection (see section
).
The technique we just described only yields convergence in capacity and
nth root asymptotics. To obtain strong asymptotics, additional assumptions must usually be made on the approximated function. This year, we proved strong asymptotics for the
aforementioned Cauchy integrals when the density of the measure is Dinismooth, and vanishes in at most finitely many points like a small fractional power. Moreover,the polar singularities of
the function are asymptotically reproduced by the approximants with their multiplicities. This result is important for inverse problem of mixed type, like those mentioned in section
, where monopolar and dipolar sources are handled simultaneously. Convergence even holds on the support of the measure if the
latter is analytic. The method is to translate the critical point equation into trigonometric orthogonality on
Twhere harmonic analysis techniques can be used. Analyzing the orthogonality in terms of Hankel and Toeplitz spectral equations, the conclusion ultimately follows from estimates on the
essential spectrum of Hankel operators. This type of result is new, for the density may vanish at some points (although in a control manner). When relaxed to multipoint interpolation schemes
instead of optimal approximation, it allows one to consider rather general arcs that need not be geodesic, which is also new. An article is being written on this topic
. Moreover, these results yield bounds on the multiplicity of the singular values of the underlying Hankel
operators
.
The sharp asymptotics mentioned above have been used to obtain uniqueness of a critical point (hence of a local
minimum) in rational
L^{2}(
T)approximation to the corresponding functions
f. The method of proof, based on the index theorem and on some estimates of the Hessian matrix, closely follows
, but appeals to
in order to bound the distance of the poles from
. This is the first uniqueness result for Cauchy integral of
complexmeasures.
Through the doctoral work of V. Lunot, a different rational approximation problem, has been taken up. More precisely, we considered the approximation of a function
fin the Schur class (
i.e.those functions analytic in
Dand bounded by 1 in modulus) by a rational function of given degree which is also in the Schur class. This problem is quite important for the modelling of passive systems. Our technique
dwells on the multipoint Schur algorithm
. After the seminal work in
, the links between the classical Schur interpolants and the orthogonal polynomials on the circle associated to the
Clark measure representing
(1 +
z
f)/(1
z
f)are well understood. When the multipoint algorithm is used on a sequence of points that remains in a compact subset of
D, corresponding generalizations were carried over with orthogonal polynomials replaced by orthogonal rational functions. However, in order to efficiently approximate functions that do
not extend analytically across
T, it is necessary to let the interpolation points tend to
T. Using weighted polynomials approximation
, we constructed a distribution of interpolation points in
Dthat produces a hyperbolic analog to the Szegö theorem. Namely, the hyperbolic distance between
fand the
nth interpolant goes to zero when integrated against the Poisson kernel at the
nth interpolation point. This result lends perspective to the use of interpolation points near the boundary in order to locally reduce the approximation error. This piece of research
will be part of V. Lunot's doctoral dissertation.
It is known after
that the denominators of best rational of meromorphic approximants to a function
fin the
L^{p}norm on a closed curve (say the unit circle
Tto fix ideas) satisfy for
p2a nonHermitian orthogonality relation when
fis the Cauchy transform of a complex measure on a curve
(the locus of singularities) contained in the unit disk
D. This has been used in the last two years to assess the asymptotic behavior of the poles of the approximants when
is a hyperbolic geodesic arc. Specifically, under weak regularity conditions on the measure, the counting measure of the poles converges weakstar to the equilibrium distribution of the
condenser
(
T,
)
,
. Let us mention that the more general situation where
is a socalled “minimal contour” for the Green potential (of which a geodesic arc is the simplest example) has been settled with the same conclusion. This technical result is still under
writing. It is of particular significance to locate several 2D sources or piecewise analytic cracks from overdetermined boundary data (see sections
and
).
Strong asymptotics were also obtained last year, that deal with the behavior of
allthe poles and not just a full proportion of them. They were established for
fthe Cauchy transform of a smooth nonvanishing complex measure on a hyperbolic geodesic arc in the disk, provided the density increases at least like a fractional power at the endpoints
of the arc. This year, using Power's theorem on the essential spectrum of Hankel operators, we were able to handle cases where the density
doesvanish. On the one hand, this vanishing is severely restricted: it may only occur in finitely many points and like a fractional power with a bound on the exponent. On the other hand,
it is known that conditions have to be put on the zeroing for such a result to hold, and it is apparently first of its kind. Although the bound is likely to be far from optimal, this sheds
light on the fundamental phenomenon of spurious poles.
The case where a rational function is added to the approximated function has also been handled, generalizing results of Gonchar and Suetin
. A numerical illustration is shown in Figures

for various approximants to the functions
Fand
Ggiven below.
Groomed by industrial users like Thales AleniaSpace, we started to analyze the realizations of 2x2 lossless scattering systems whose scattering response
(
S
_{i,
j})satisfies the socalled
selfreciprocalcondition
S_{1, 1}=
S_{2, 2}. In the filterdesign community, it was common belief that every selfreciprocal response can be realized in the form of an electrical circuit with plane symmetry. In
short, such circuits should be identical when viewed from each access. Whereas it is clear that symmetric circuits have selfreciprocal responses, we showed this year that the converse does not
hold. More precisely the algebraic variety of autoreciprocal responses decomposes into several prime components. Only one of them consists of systems that can be realized by symmetric
circuits. In particular we gave examples of some dual band responses that have no symmetric physical realizations whatsoever. In addition, we connected the prime decomposition of the variety
with an interlacing property of the transmission and reflexion zeros. This yields an “almost visual” procedure to decide whether a symmetric network can realize a given frequency response.
In future work, we will focus on the practical implementation of this analysis within the software DedaleHf .
A general survey on applications of our work to microwave filters synthesis has been published .
This piece of research is part of V. Lunot's doctoral work. The theoretical developments took place over the last two years, while deepenings of the numerical aspects were carried out in 2007. The problem goes as follows. On introducing the ratio of the transmission and reflexion entries of a scattering matrix, the design of a multiband filter response (see section ) reduces to the following optimization problem of Zolotarev type :
where
(resp.
) is a finite union of compact intervals
I_{i}of the real line corresponding to the passbands (resp. stopbands), and
P_{m}(
K)stands for the set of polynomials of degree less than
mwith coefficients in the field
K. Depending on the physical symmetry of the filter, it is interesting to solve problem (
) either for
(“real” problem) or
(“mixed” problem), or else
(“complex” problem). The “real” Zolotarev problem can be decomposed into a sequence of concave maximization problems, whose solution we were able to characterize in terms of an
alternation property. Based one this, a Remezlike algorithm has been derived in the polynomial case (
i.e.when the denominator
qof the scattering matrix is fixed), which allows for the computation of a dualband response (see Figure
) according to the frequency specifications (see Figure
for an example from the spacecraft SPOT5 (CNES)). We have designed an algorithm in the rational case was well which, unlike
linear programming, avoids sampling over all frequencies. This raises the issue of the “generic normality” (
i.e.the maximum degree) of the approximant with respect to the geometry of the intervals. This question remains open. The design of efficient procedures to tackle the “mixed” and
“complex” cases remains a challenge. A preliminary version of a software by V. Lunot to treat the complex case has been released this year to our academic partners: Xlim and the Royal Military
College of Canada. Applications of the Remez algorithm to filter synthesis are described in
,
. An article on the general approach based on linear programming has been accepted for publication
.
A collaboration has started with the Royal Military College of Canada on the modeling and synthesis of broad band microwave filters. Until now, the team's work on microwave filters has focused on narrowband devices where the ratio between the passband and the middle frequency is about 1%. This allows the use of lowpass transformations leading to abstract equivalent circuits containing nonphysical elements, like impedance inverters ( i.e.frequencyindependent coupling) Recent applications in the GSM sector calls however for the use of filters with broader passband. Whereas the lowpass formalism clearly breaks down for relative passbands above 5%, the expansion of the filter's response over finitely many modes of the Helmholtz equation can still be done, justifying the rational character of the response in his case too. We showed, that using a modified version of Zolotarev's problem with nonpolynomial weight, it is again possible to synthesize optimal responses in the high frequency domain. Moreover an equivalent circuit formalism involving “physical”capacitive couplings was developed in order to help the actual realization of the device. The tuning requires a reliable version of the PrestoHf software in the high frequency domain. In this new setting where no lowpass transformation can be made, challenging numerical problems arising from the relative smallness of the approximation band in the underlying bounded extremal problems must be overcome. This will be a major research effort of the team in the future. Preliminary results on the realization of suspended strip line filters with relative band up to 30%(in collaboration with the university of Ulm, Germany) are encouraging.
Another pending issue is that of controlling the group delay, that is, the derivative of the phase of
p/
q. The latter becomes large around abrupt changes of the modulus, and an approach based on perturbing the “optimal” poles to keep the logarithmic derivative more flat
is currently being investigated.
An OMUX (Output MUltipleXor) can be modeled in the frequency domain through scattering matrices of filters, like those described in section , connected in parallel onto a common guide. The problem of designing an OMUX with specified performance in a given frequency range naturally translates into a set of constraints on the values of the scattering matrices and of the phase shift introduced by the guide in the considered bandwidth.
An OMUX simulator on a Matlab platform was designed in recent years and used to check various assumptions about how the OMUX operates; in particular, that each rightsection of the OMUX (when the guide gets oriented from the left to the right with common access to the extreme left) acts as a shortcircuit to those channels lying “upstream” in their bandwidth. Another assumption was that each channel must reflect a certain amount of energy in its bandwidth in order to trap waves otherwise reflected by the abovementioned shortcircuit. Under the terms of a contract with Alcatel Alenia Space (see section ), these assumptions have been used to design a dedicated software to optimize OMUXes whose first version has just been released to AAS.
The direct approach, used up to now by the manufacturer, consists in coupling a simulator with a general purpose “optimizer”, in order to reduce transmission and reflection wherever they are too large. This does not perform well for high degree and narrow bandwidth, where convergence often fails and multiple initial points must be used, resulting in a lengthy and sometimes unsuccessful search. The software proceeds by adding channels recursively, applying to the new channel the above shortcircuit and reflectioninthe bandwidth rules. This yields an initial guess for the global “optimizer” which seems to regularly outperform those currently used by AAS. More extensive tests are being conducted. The problem of “manifoldpeaks” has not been improved yet, and would require a dedicated study.
No new result was obtained this year on this topic, but the article has appeared. It mostly reports on results from David Avanessoff's PhD. The subject is the dynamic linearizability, or parametrizability (see section ) of control systems with 3 states and 2 controls; this essentially covers the same range of systems as those with 4 states and 3 controls that are affine with respect to the control.
Such systems were previously studied in
where it is shown they can be “
(
x,
u)flat” only if they admit a parametrization of order at most (1,2),
i.e.one parametrizing function is differentiated at most twice and the other once. We conjecture that other systems cannot be parametrized at all (
i.e.for no order). What we prove in
is that they do not admit parametrizations of order (3,3) nor of order
(2,
k)for any
k.
This result is stronger than those in and also much more readable, as the latter reference used computer algebra in the proofs.
This research has been initiated with the postdoctoral stay of H. Jirari, and its goal was described in section . The systems under study here are quantum systems of low dimension (1 or 2 Qbits) whose autonomous dynamics is given by the Schrödinger equation and whose interaction with the environment is described by some dissipative term. According to the latter, one distinguishes between the BlochRedfeld model and the (heavily simplified) Lindblad model. In , numerical evidence was obtained that for the BlochRedfeld model, at least in simple small dimensional cases, there exists a control that drives the system from some pure initial state to a target one, while compensating the dissipative effect of the environment. We have pursued this numerical effort on more strongly coupled environments, computing viaoptimal control techniques. We also begun the investigation of the controltheoretic structure of such models. In contrast with the Lindblad model, where decoherence is unavoidable in some cases, it seems that the BlochRedfeld model exhibits a more robust behavior which calls for further understanding.
Most of this section accounts for results obtained in Alex Bombrun's PhD work; the thesis was defended on March 12, see .
It deals with the orbit transfer of satellites equipped with low thrust engines like plasmic ones, which are energyefficient but deliver a much smaller thrust than conventional “chemical” engines: the ratio between acceleration and gravity is like 10 ^{3}, sometimes less. This problem was raised by Alcatel Alenia Space, who supports our research effort on the subject (see section ).
Low thrust satellites are prototypical of conservative systems with small controls,
i.e.control systems whose drift is a conservative dynamical system having one or more first integrals, and whose control is constrained to be very small. Here, the drift is the Kepler
system, a completely integrable Hamiltonian system with five independent first integral in a sixdimensional phase space. In the terminology of
, a
Kepler control systemis a system in dimension
nwhose drift has
n1first integral and compact trajectories (for the satellite, this is true only for negative energy, in the socalled elliptic domain) and where the control is small
in the sense of certain asymptotic properties as the bound on the control tends to zero.
The first results concern the computation of feedback controls, using
ad hocLyapunov functions.These approximate surprisingly well timeoptimal trajectories. The easy implementation of such control laws makes them attractive as compared to genuine optimal
control. Here the
n1first integrals are an easy means to build control Lyapunov functions since any function of these first integral can be made monotone decreasing by a suitable
control. The surprising result is that fitting a Lyapunov function as close as possible to
one particulartimeminimum trajectory yields a feedback control that performs excellently on a wide range of initial conditions. This is under writing for publication. To make the study
more systematic, we started an asymptotic analysis when the thrust tends to zero. Control Lyapunov functions can then be tested on the average control system described in section
; this research is still in course.
For the class of Kepler control systemsthat we just defined in section , a notion of average control systemis introduced in . Using averaging techniques in this context is rather natural, since the free system produces a fast periodic motion and the smallcontrol a slow one; averaging is a widespread tool in perturbations of integrable Hamiltonian systems, and the small control is in some sense a “perturbation”. In some recent literature, one proceeds as follows: the control is preassigned, for instance to time optimal control via Pontryagin's Maximum Principle or else to some feedback designed beforehand. Then, averaging is performed on the resulting ordinary differential equation, whose limit behavior is analyzed when the control magnitude tends to zero.
The novelty of (see also ) is to average beforeassigning the control, hence getting a control systemthat describes the limit behavior better. For that reason, the average control system is a convenient tool when comparing different control strategies.
It allowed us to answer an open question stated in on the minimum transfertime between two elliptic orbit when the thrust magnitude tends to zero, see .
Under some controllability conditions that are trivially satisfied in the case at hand, we proved that the average system is one where the velocity set has nonempty interior, i.e. all velocity directions are allowed at any point, and the constraint is convex; mathematically this yields a Finsler structure (in the same way as a controllable system without drift with a quadratic constraint on the control yields a subRiemanian structure). An article is in progress on these results; the latter can be found in .
In 1955, Serre conjectured that every row vector with coefficients in a commutative polynomial ring
A=
k[
x_{1}, ...,
x_{n}]over a field
kwhich admits a rightinverse over
Acan be completed into a square matrix with a nonzero constant determinant. This conjecture was independently proved by Quillen and Suslin in 1976 and is nowadays called the
QuillenSuslin theorem. Within module theory, the QuillenSuslin theorem means that a projective
Amodule is free and the computation of the unimodular matrix gives an explicit basis of the free module. Constructive proofs of the QuillenSuslin theorem have largely been studied in
the symbolic computation literature but no implementation was available till now. A Maple implementation of the QuillenSuslin theorem called
QuillenSuslinhas recently been achieved by A. Fabiańska (University of Aachen, Germany) in collaboration with A. Quadrat and is
demonstrated in
,
. Based on the QuillenSuslin theorem, we solve in
,
open problems in mathematical systems theory (e.g., LinBose's conjectures, computation of (weakly) left/right
coprime factorizations of multivariate rational transfer matrices, computation of flat outputs and injective parametrizations of flat differential timedelay systems, LieBäcklund equivalence
of flat differential timedelay linear systems). The corresponding algorithms have been implemented by the authors in the package
QuillenSuslin.
A wellknown but difficult result in noncommutative algebra due to J. T. Stafford asserts that every projective left module
Mover the Weyl algebras
D=
A_{n}(
k)or
B_{n}(
k)of differential operators with polynomial/rational coefficients (
kis a field of characteristic 0) with
rank
_{D}(
M)
2is free. The purpose of
is to present a constructive proof of this result as well as an effective algorithm for the computation of bases
of
M. This new algorithm has been implemented by the authors in the package
Staffordof the library
OreModules
. The main motivation for studying this open algorithmic problem is to effectively compute injective
parametrizations and flat outputs of flat underdetermined linear systems of partial differential equations (Monge problem)
.
The factorization and decomposition problems of linear systems of ordinary differential or difference equations have long been studied in the mathematical and symbolic computation literature. The main novelty of the approach developed in is to recast these problems within the framework of algebraic analysis and to extend them to general linear systems defined over certain classes of functional operators called Ore algebras (e.g., partial differential equations, differential timedelay equations, recurrence equations). In particular, gives general conditions for a linear functional system to admit nontrivial factorizations and decompositions. Under certain conditions (free modules), we prove that the system is equivalent to a block triangular or a block diagonal system. The different algorithms have been implemented in the package Morphismsof OreModules . Using the packages Staffordand QuillenSuslin, Morphismsgives us a constructive way to study the factorization and decomposition problems for general linear functional systems. A library of examples demonstrates the package Morphismson examples coming from mathematical physics and control theory.
Contract (reference INRIA: 2470, CNES: 60465/00) n ^{o}04/CNES/1728/00DCT094 involving CNES, XLIM and INRIA, whose objective is to work out a software package for identification and design of microwave devices. The work at INRIA concerns the design of multiband filters with constraints on the group delay. The problem is to control the logarithmic derivative of the modulus of a rational function, while meeting specifications on its modulus.
A contract (reference INRIA: 1931, AAS: B00375) has been signed between INRIA and Alcatel Alenia Space (branch of Toulouse), in which INRIA will design and provide a software for OMUX simulation with efficient initial condition for an optimisation algorithm based on recursive tuning of the channels.
The aim of this contract (reference INRIA: 2267, AAS: B02173) was to demonstrate the feasability of the approach taken in A. Bombrun's PhD for lowthrust satellite orbit transfer. A Lyapunovfunctionbased design method has been successfully implemented in a numerical Matlab code and validated on real transfer situations.
L. Baratchart is a member of the editorial board of Computational Methods and Function Theoryand Complex Analysis and Operator Theory.
J. Leblond is a member of the scientific committee of PICOF'08 (4th International Conference on Inverse Problems, Control and Shape Optimization).
A. Quadrat is an associate editor of the international journal Multidimensional Systems and Signal Processing(Springer).
The team was awarded two ANR grants in 2007.
The first one, entitled AHPI (Analyse Harmonique et Problèmes Inverses), is a “Projet blanc” in Mathematics involving INRIASophia (APICS, coordinator), the Université de Provence (LATP), the Université Bordeaux I (LATN), the Université d'Orléans (MAPMO), INRIAFuturs (Magic 3D), the Université de Pau. The coordinator is L. Baratchart. It aims at developing Harmonic Analysis techniques to approach inverse problems in seismology, Electroencephalography, tomography and nondestructive control.
The second one, entitled FILIPIX, is a “Projet Thématique en Télécommunications”, involving INRIASophia (APICS), XLIM, AAS (Centre de Toulouse, coordinator).
The project is a member of the Working Group Control and System Theory of the
ERCIMconsortium, see
http://
A collaboration entitled Computational Methods in Mathematical Systems Theorybetween the Apics project and the Lehrstuhl B für Mathematik RWTH  Aachen was supported by the grant PAI Procope. Within this framework, we have received the visits of Daniel Robertz, Anna Fabiańska and Mohamed Barakat.
EPSRCresearch grant EP/F020341/1 (Operator theory in function spaces on finitelyconnected domains), with Leeds University (UK) and the University Lyon I, 20072009.
INRIADGRSTUniv. tunisiennes(STIC) with LAMSINENIT (Tunis), « Problèmes inverses du Laplacien et approximation constructive des fonctions » (from which M. Zghal received financial support for her PhD). Apics est l'une des EPI partenaires (avec Anubis et Poems) du LAMSIN au sein de l'équipe associée Enée.
NSF EMS21RTG students exchange program (with Vanderbilt University).
The following scientists gave a talk at the seminar:
Andrei Agrachev, SISSA Trieste, Acad des Sciences Moscou, Geometry of Rolling Bodies and Optimal Control.
Silvère Bonnabel, CAS, ENSMP, Observateurs et Symmétries.
Mohamed S. Boudellioua, Sultan Qaboos University, Oman, Equivalence and Factorization of Multivariate Polynomial Matrices.
José Grimm, Putting math on the Web with Tralics.
Amadeo Irigoyen, Équipe d'analyse complexe, Univ. Paris VI, Quelques applications de la théorie d'approximation nonlinéaire en théorie inverse de SturmLiouville.
Ian Jermyn, Inria Ariana, Contours actifs d'ordre supérieur et champs de phase pour la compréhension d'image.
Hamza Jirari, Contrôle optimal en mécanique quantique.
Vincent Lunot, Un problème de Zolotarev. Application à la synthèse de filtres multibandes.
Alban Quadrat, Factorisation et décomposition des systèmes linéaires fonctionnels.
Thomas Ransford, Université de Laval, Canada, Pseudospectres et croissance de puissances.
Eva Sincich, RICAM, Austrian Academy of Sciences, Linz, Stability for some inverse scattering problems.
L. Baratchart, DEA Géométrie et Analyse, LATPCMI, University Marseille I.
J. Leblond, Centre Montessori, collège, MouansSartoux.
M. Olivi, Mathématiques pour l'ingénieur (Fourier analysis and integration), section Mathématiques Appliquées et Modélisation, 3ème année, Ecole Polytechnique NiceSophia Antipolis.
A. Quadrat, ISIA, Master affiliated to the Ecole des Mines de Paris (computer algebra).
Alex Bombrun, « Les Transferts Orbitaux à Faible Poussée : Optimalité et Feedback » (optimal control, feedback, and orbital transfert for low thrust satellite orbit transfer). Defended March 12, 2007.
Imen Fellah, « Complétion de données dans les espaces de Hardy et problèmes inverses pour le Laplacien en 2D », cotutelle with LamsinENIT (Tunis), .
Vincent Lunot, « Problèmes fréquentiels extrémaux, approximation rationnelle sous contrainte Schur et application à la synthèse de filtres ».
Moncef Mahjoub, « Approximation harmonique dans une couronne et applications à la résolution numérique de quelques problèmes inverses », cotutelle with LamsinENIT (Tunis).
Meriem Zghal, « Constructive aspects of some inverse problems (Cauchy, sources) for Laplace equation in ellipsoidal domains », cotutelle with LamsinENIT (Tunis).
L. Baratchart was on the PhD defense committee of Philippe Jaming, Univ. Orléans.
L. Baratchart was on the “soutenance d'habilitation” committee of Maureen Clerc, Univ. Nice.
J. Leblond was on the PhD defense committee of Imen Fellah, LamsinEnit, Tunis.
M. Olivi was on the PhD defense committee of Tom D'haene, Vrije Universiteit Brussel.
L. Baratchart is INRIA's representative at the « conseil scientifique » of the Université de Provence.
J. Grimm is a representative at the « comité de centre ».
J. Leblond is a member of the « Commission d'évaluation » of INRIA.She is in charge with the Séminaires Croisés of the Research Centre, participates to the working group « Méditerranée 3+3 » and has participated to the working group « Accueil des stagiaires étrangers ».
M. Olivi is a member of the CSD (Comité de Suivi Doctoral) of the Research Unit of Sophia Antipolis.
A. Quadrat was a member of the Jury Concours CR2, INRIA Futurs, Lille 2007.
J.B. Pomet is a representative at the « comité technique paritaire » (CTP).
F. Seyfert is a member of the CDL (Comité de Développement Logiciel) of the Research Unit of Sophia Antipolis.
L. Baratchart, J. Grimm, J. Leblond and M. Olivi attended the 2007 ERNSI Meeting in San Servolo (Italy), where M. Olivi presented a talk on “Real Symmetric Darlington Synthesis.”
V. Lunot gave a talk at the International Microwave Symposium “Optimal Synthesis for MultiBand Microwave Filters”, Honolulu, U.S.A, June
J. Leblond and S. Rigat attended the International Conference in Mathematics in honor of G. Henkin, IHP, Paris, June.
J. Leblond attended the forum on information and communication science and technologies, to celebrate 40 years of activity at INRIA, Lille, December.
L. Baratchart was an invited speaker at the colloquium of the Université de Bourgogne, Dijon, May.
L. Baratchart was an invited speaker at the Workshop “Modern Approaches in Orthogonal Polynomials”, Banff, Alberta, CN., November.
L. Baratchart, J. Leblond and S. Rigat gave communications at the workshop “ Analyse microlocale et harmonique pour les problèmes inverses ”, CIRM, Marseille, March.
L. Baratchart was an invited speaker and S. Rigat gave a communication at Mathestia, Bidart, April.
L. Baratchart gave a communication at the conference “Special functions, information theory and Mathematical Physics”, Granada, SP., September.
L. Baratchart and S. Rigat participated while J. Leblond gave a communication at the meeting of the ANR project AHPI, Pau, October.
J. Leblond was an invited speaker at MEMO, a conference on numerical methods and modelling to celebrate the 10th birthday of LAMSIN, Tunis, December. December.
J.B. Pomet gave a talk at the conference in honor of Claude Lobry, St Louis, (Senegal).
J.B. Pomet was an invited speaker at the 80 ^{th}RCP meeting on interactions between math and Physics, Strasbourg (France).
A. Quadrat gave two talks at the Journées Nationales de Calcul Formel, Luminy (France), 29/0102/02.
A. Quadrat gave a talk at the working group EDPMOSAR of the GDR MACS, ENST (Paris), 2323/03.
A. Quadrat was an invited speaker at the Journées DoctoralsJournées Nationales du GDR MACS, Reims (France), 0911/07.
A. Quadrat was an invited speaker at the Worskop on Control Distributed Parameter Systems (CDPS 2007), Namur (Belgium), 2327/07, in the session dedicated to the tribute to F. Callier's scientific career.
A. Quadrat gave two talks at the Journée “Intéractions entre théorie algébrique et calcul scientifique: Etat de l'art et applications”, CNAM (Paris), 18/09.
A. Quadrat gave a talk at the Journées de l'ANR GECKO, INRIA Sophia Antipolis, 1921/11.
Concerning their joint work with Apics, our collaborators took the following actions:
A. Gombani gave a communication at LynSys2007, Canberra, February.
M. Mahjoub gave a communication at the Mediterranean Conference on Biomathematics, Cairo, Egypt, June.
J.R. Partington gave two talks at the regular seminars of analysis, at Glasgow University (Feb.) and at King's College in London (Nov.).
R. Peeters gave a communication at the "Benelux meeting on systems and control" (Lommel, Belgium, March).