*Systems of functional equations* or simply *functional
systems* are systems whose unknowns are functions, such as systems
of ordinary (OD) or partial differential (PD) equations, of differential time-delay
equations, of difference equations, of integro-differential equations,
etc. , .
Functional systems play a fundamental role in the mathematical
modeling of physical phenomena studied in natural science such as
physics, or in engineering sciences such as mathematical systems theory
control theory, signal processing, etc. , . Numerical aspects of functional systems, especially OD and PD
systems, have largely been studied in applied mathematics due to the importance of
numerical simulation issues.

Complementary approaches, based on algebraic and differential or algebraic geometric methods, are usually upstream or help the numerical simulation of systems of functional systems. These methods tackle questions and problems such as algebraic preconditioning, elimination and simplification, completion to formal integrability or involution, computation of integrability conditions or compatibility conditions, index reduction, reduction of variables, choice of adapted coordinate systems based on symmetries, computation of first integrals of motion, conservation laws, and Lax pairs, study of Liouville integrability or of the asymptotic behavior of solutions at a singularity, etc. For more details, see , , , , , , , , , , and the references therein.

Let us state a few interests of an algebraic approach for the study of functional systems:

Algebraic methods are clearly suitable for an algorithmic study, and thus for the development of efficient algorithms implementable in computer algebra systems.

It can be used to finely study the behavior
of the solutions of a system with respect to unfixed model
parameters

The existence of closed-form solutions can highly simplify certain problems studied in applications by avoiding the use of time-consuming optimization problems, and thus fits well with nowadays real-time applications.

The GAIA team aims to develop algebraic and geometric methods for the study of functional systems.

Although not yet very popular in applied mathematics, algebraic and differential geometric approaches of functional systems have lengthly been studied in foundamental mathematics. We can state a few names of mathematical theories such as (differential) Galois theory, Lie groups, exterior differential systems, differential algebra, algebraic analysis, etc. , , , , , , , , , , .

Over the past years, some of these algebraic theories for the study of functional systems have been investigated in the computer algebra community within an algorithmic viewpoint, mostly driven by applications to engineering sciences such as mathematical systems theory and control theory.

Gröbner or Janet bases ,
for noncommutative polynomial
rings of functional operators or differential elimination
techniques for differential
systems , , ,
based on differential algebra , ,
are remarkable examples of those effective algebraic methods. They
are nowadays implemented in standard computer algebra systems (e.g. `Maple`, `Mathematica`, `Magma`).

These effective algebraic approaches also form the algorithmic “engines” at the basis of the first developments of effective versions of modern algebraic theories (algebraic geometry, differential algebra, module theory and homological algebra over certain noncommutative polynomial rings of functional operators, algebraic analysis, etc.).

The above-mentioned results are just the tip of the iceberg and much more effort must be made in the future for making effective larger parts of fundamental mathematics and making them largely available in standard computer algebra systems. This “democratization” process towards the accessibility of fundamental mathematics is important, for instance, for educational issues where computers can be used to teach them to students, scientists of other communities and engineerings, and for learning by doing and computing. Further developing effective mathematics, making them accessible to a larger audience through dedicated software, and demonstrating them through interesting engineering problems are at the core of the GAIA team. The latter engineering problems are major sources of motivation for the development of effective algebraic theories and their implementations in computer algebra systems as explained in the next section.

A major source of motivation for the development of the effective study of algebraic theories is represented by control theory issues. Indeed, certain problems studied in control theory can be better understood and finely studied by means of algebraic or geometric structures and techniques. The rich interplay between algebra, computer algebra, and control theory has a long history.

The first main paper on Gröbner bases written by their creators, Buchberger, was published in Bose's book on control theory of multidimensional systems since they play a fundamental role in this theory. They were the first main applications of Gröbner bases outside the field of algebraic geometry and they still play a fundamental role in multidimensional systems theory .

The differential algebra approach to nonlinear control theory , , ,
was a major motivation for the effective study of differential algebra ,
(differential elimination theory, triangular sets, regular chains, etc.)
, , and its implementations in `Maple`. Within this effective differential algebra approach to
nonlinear control systems, observability, identifiability, parameter
estimation, invertibility, differential flatness,
etc., have received appealing and checkable algebraic characterizations.

Linear control theory
and multidimensional systems theory ,
have recently been profoundly developed due to the so-called behavior
approach ,
and the module approach , .
Based on ideas of *algebraic analysis* ,
system properties of those systems (e.g. controllability,
parametrizability, differential flatness) are intrinsically characterized by means of properties of
certain algebraic structures (namely finitely presented left modules over
noncommutative polynomial rings of functional operators). To effectively check the latter
properties, the development of effective versions of
two important algebraic theories, namely module theory
and homological algebra , had to be iniated
based on functional elimination techniques (i.e. Gröbner or Janet
basis techniques for noncommutative polynomial rings) , (see also ).
Dedicated packages, written in `Maple`, `Mathematica` and `GAP`, are now available.

The GAIA team wants to further develop its expertise in this direction by considering new classes of functional systems (e.g. differential varying/distributed delay systems, ordinary integro-differential systems) interesting in control theory and in signal processing.

The first goal of the GAIA team is to study classes of
*functional systems* which are interesting in practice (e.g. differential systems, differential constant/varying/distributed delay systems, ordinary integro-differential systems) by means of *algebraic* and *geometric*
methods (algebraic analysis, algebraic/differential/noncommutative geometry, etc.),
*computer algebra* (e.g. algorithmic, symbolic and symbolic-numeric
methods, librairies), and *mathematical systems theory*. The systems to be investigated can be
linear, nonlinear, continuous, discrete, or originated
from real life applications.

The second goal of the GAIA team is to study important problems coming from:

*control theory* (e.g. parametric robust control, stability and
stabilization of multidimensional systems or of differential constant/distributed/time-varying delay systems)

*signal processing* (e.g. parameter estimation problem,
metric multidimensional unfolding, autocalibration)

*multidisciplinary domains* (e.g. marine bivalves
behavior, human-machine interaction, chemical reaction networks, ionic activities in
neuroscience)

The third goal of the GAIA team is to develop (`Maple`, `Mathematica`, `C`/`C`++) *packages and librairies*
dedicated to functional systems and to their applications, and in parallel,
eventual *industry transfer* (e.g. *Safran Electronics & Defense*, *Safran Tech*, *Maplesoft*).

To develop a computational study of problems coming from control theory, signal processing, and multidisciplinary domains, parts of algebraic theories must be studied within an effective approach: methods and theoretical results must be made algorithmic based on computer algebra techniques appropriated for efficient implementations in computer algebra systems.

The works on stability and stabilization
problems of multidimensional systems, developed in the former ANR
MSDOS (2014–2018),
have shown the importance for developing an effective version of the
module theory over the ring of rational functions without poles in
the closed unit polydisc of *Polydisc Nullstellensatz* , a natural extension of Hilbert's *Nullstellensatz* for the above-mentioned ring (see e.g. ). In addition,
the existence of a doubly coprime factorization is related to a
theorem obtained by (the Fields medalist) Deligne with a
non-constructive proof . This theorem can be seen as an
extension of the famous Quillen-Suslin theorem (Serre's conjecture) .
Based on our experience of the first implementation of the
Quillen-Suslin theorem in the computer algebra system (`Maple`)
, we aim to develop this effective framework as well as a
dedicated `Maple` package.

A differential geometric counterpart of differential algebra and differential elimination theory , is the so-called Spencer's theory of formal integrability and involutive PD systems , . For linear PD systems, this theory can be seen as an intrinsic approach to Janet or Gröbner bases for noncommutatibe polynomial rings of PD operators. No complete algorithmic study of Spencer's theory has been developed yet. We aim to develop it as well as to implement it. The understanding of the connections between the different differential elimination theories (Janet or Gröbner bases , Thomas decomposition , differential algebra , , Spencer's theory , exterior differential systems , etc.) will also be investigated. On a longer term, applications of Spencer's theory to Lie pseudogroups and their applications in mathematical physics (e.g. variational formulations based on Lie (pseudo)groups) will be investigated and implemented.

The main contribution of this axis is the development of effective elimination theories for both linear and nonlinear systems of integro-differential equations.

The rings of integro-differential operators are more complex than the purely
differential case , due to the existence of
zero-divisors or the fact of having a *coherent ring* instead of a
*Noetherian ring* . We want to
develop an algorithmic study of these rings. Following the direction
initiated in for the computation of zero divisors,
we first want to develop algorithms for the computation of
left/right kernels and left/right/generalized inverses of matrices
with entries in such rings, and to use them to develop a module-theoretic approach to linear systems of integro-differential
equations. Following ,
standard questions addressed within the computer algebra community
such as the computation of
rational/exponential/hyperexponential/etc. solutions will also be
addressed. Moreover, famous Stafford's results ,
algorithmically studied in , for rings of
PD operators, are known to still hold for rings of
integro-differential operators . Their algorithmic
extensions will be investigated and our corresponding implementation will be
extended accordingly. Finally, following , , an algorithmically study of rings
of integro-differential-delay operators will be further developed
as well as their applications to the equivalence problem of
differential constant/varying/distributed delay systems
(e.g. Artstein's reduction, Fiagbedzi-Pearson's transformation) and
their applications to control theory.

*Integro-differential algebra* is an extension of Ritt-Kolchin's *differential algebra* ,
that also includes integral operators. This extension is now attracting more attention in mathematics, computer algebras, and control theory.
This new type of algebras will be algorithmically studied for integro-differential
nonlinear systems. To do that, concepts such as integro-differential
ideals and varieties have to be introduced and studied for developing an integro-differential elimination
theory which extends the current differential elimination theory , , . A
`Maple` prototype will first be developed and then a `C` library
when experience will be gained.

We aim to further reinforce our expertise in the computer algebra aspects of functional systems and algebraic curves by attacking remaining technical obstacles and by considering new classes of functional systems, notably those coming from interesting applications in engineering sciences and particularly in control theory.

On an algorithmic viewpoint, there are mainly four different approaches for the study of the singularities of plane algebraic curves. Let us shortly list them:

The well-known *Newton-Puiseux* algorithm, initiated by Cramer and Puiseux, which follows an idea due to Newton. This
approach has successively been improved in , , , .

The *Extended Hensel Construction*, developed in
and recently improved in .

The first two methods are based on Puiseux series computations. They
use techniques which are equivalent to the standard blowing-up of a
singularity of an algebraic curve, which has the drawback to be
bottlenecks in terms of complexity and practical
efficiency. Nevertheless, the recent work
provides the best complexity currently known.
The last two methods study singularities without computing Puiseux
series. They both use the concept of an *extended evaluation*.

A recent very efficient algorithm for the factorisation of a univariate polynomial based on its Newton polygon has recently been obtained in . The key ingredient of this algorithm is to work on the given polynomial and not after changes of variables as usually done in the literature.

To improve the complexity results of , , we want to combine the above different approaches using approximate roots and a generalization of the results of to the context of .

The method proposed above is important in
practice since it is based on well-known and efficiently
implemented algorithms (mainly Newton iteration and gcd computations). Nevertheless, these algorithms involve technical difficulties
on the computer science side: the main one is the need to
improve the accuracy of computations due to truncations. These issues, including also run-time compilation, are well-studied in
the `BPAS` library

Another part of the code development concerns issues on certified numerical computations for univariate polynomials (with algebraic coefficients) that will be used for the development of certified symbolic-numeric algorithms making effective the strategy proposed in .

A major bottleneck of computational differential algebra
methods is the computation of greatest common divisors of multivariate
commutative polynomials. Any algorithmic progress in this direction would highly improve the efficiency of differential
algebra software such as, for instance, the `C` library `BLAD`
.
Moreover, numerous computer algebra problems and related implementations could also highly profit from any success in this direction.

A major application of the effective differential algebra approach developed by GAIA's members , is the possibility to reduce a nonlinear (implicit) differential system, particularly differential algebraic equations, to so-called regular chains of differentiation index 0, i.e. to systems which do not need differentiation of their equations to be rewritten as pure differential systems . Based on our expertise on differential techniques, we want to study the consistent initialization problem and develop numerical integrators for nonlinear differential algebraic systems, and used them in the study of coupled algebraic and differential systems, interconnected systems, or networks .

A package dedicated to nonlinear integro-differential equations will be developed in a `Maple` prototype and then in a standalone `C`/`C`++ open source library (as it was already done for the `diffalg` and `DifferentialAlgebra` packages). General purpose solvers such as `Maple` `dsolve` or `pdesolve` may call differential elimination methods for computing essential singular solutions of differential equations, for computing systems of polynomial differential equations admitting a given function as a solution, etc. On the long run, one may foresee enhanced general purpose solvers able to handle integro-differential equations processed through integro-differential elimination methods. It will rely on a sub-package dedicated to the problem of effectively handling integro-differential expressions.

These packages will rely on existing software such as `BLAD`, `DifferentialAlgebra`, and `MABSys`. It is worth pointing out that proof-of-concept methods are
already available. See and .
The collaboration with modelers will also enhance the software user-interface for
a better usability.

The study of numerical integration of integro-differential equations
(a necessary component of software dedicated to the parameter
estimation problem) will
also be further studied following the direction initiated in ,
leading to the `Maple/C` library
`BLINEIDE`.
This software has currently no widely available challenger.

The `Maple` prototype software dedicated to nonlinear
integro-differential equations will also be implemented in a
standalone `C`/`C`++ open source library, leading to software
easier to integrate in modeling platforms such as `OpenModelica`. The GAIA team has quite some expertise in
releasing software satisfying industrial standards: its `C` open
source `BLAD` libraries, dedicated to differential elimination,
are currently integrated in `Maple` and called through the `Maple` package `DifferentialAlgebra`. The `Modelica`
programming language, which emphasizes programming with equations and
permits to call external code, can integrate software dedicated to
integro-differential equations developed in the GAIA team.

Our expertise in the computer algebra aspects to stability and stabilization problems for multidimensional systems and for differential constant/distributed/varying delay systems , , , will further be developed.

We shall investigate the possibility to develop a computer algebra
package for the design of Lyapunov functions for homogeneous dynamical
systems based on an effective study of the differential algebra of
*generalized forms*, i.e. of Puiseux polynomials in signed powers
, .

The symbolic-numeric study of the robust stability of a differential
constant time-delay system with respect to the delay `C` library. They will be used to study how the
different branches of a quasipolynomial at a critical pair (namely

Moreover, in collaboration with Mouze (Centrale Lille, France), we
want to develop an effective study of the ring

We want to use the results developed
on the module-theoretic aspects of the ring of multivariate rational
functions without poles in the closed unit polydisc of `Maple` package.

Finally, the *noncommutative geometric approach* to robust problems for infinite-dimensional linear systems (e.g. differential time-delay or PD systems) , initiated in , will be further studied based on the mathematical concepts and methods introduced by (the Fields medalist) Connes . We particularly want to investigate generalizations of Nyquist's theorem to infinite-dimensional systems
based on index theory (pairing of

Our expertise on algebraic parameter estimation problem, coming from the former Non-A project-team, will be further developed. Following , this problem consists in estimating a set *operational calculus* (Laplace transform, convolution),
shows how

A first aim in this direction is to develop to a greater extent our recent work that shows how the above approach can cover wider classes of signals such as *holonomic
signals* (e.g. signals decomposed into orthogonal polynomial
bases, special functions, possibly wavelets).

Moreover, explains how larger classes of structured
perturbations *annihilators*, is
replaced by a new approach based on the more general algebraic concept
of *syzygies* . This general approach to
the algebraic parameter estimation problem will be
developed. Following the ideas of , an
effective version of this general approach will also be done based on differential elimination techniques, i.e. Gröbner basis techniques for rings of differential operators. It will be
implemented in a dedicated `Maple` package which will extend the current prototype `NonA` package .

Furthermore, as
an alternative to passing forth and backwards from the time domain
to the operational (Laplace/frequency) domain by means of Laplace transform and its
inverse as done in the standard algebraic parameter
estimation method , , we aim to develop a direct time domain approach
based on calculus on rings of integro-differential operators as
described by the following picture (

The direct computation will be handled by means of the effective methods of rings of integro-differential operators described in the above sections.

For nonlinear control systems, the approach to the parameter estimation problem, recently proposed in and based on the computation of integro-differential input-output equations, will be further developed based on the integration of fractions . Such a representation better suits a numerical estimation of the parameters as shown in .

In , we have recently initiated an extension of the results developed in to handle integro-differential equations such as Volterra-Kostitzin's equation. This general approach, based on an extension of the input-output ideal method for ordinary differential equations to the integro-differential ones, will be further developed based on the effective elimination theory for systems of integro-differential equations. An important advantage of this approach is that not only it solves the identifiability theoretical question but it also prepares a further parameter estimation step .

We have developed a collaboration with *Safran Electronics & Defense*
(Massy Palaiseau) and Rouillier (Ouragan, Inria Paris) on a *parametric robust control theory* based on computer
algebra methods (symbolic-numeric methods), as well as its
applications to the robust stabilization of certain mechanical
systems (e.g. gyrostabilized systems, two mass-spring-damper system,
stabilized mirrors).

For low-dimensional systems of ODEs, this approach aims to determine closed-form solutions for robust controllers and for the robustness margins in terms of the model parameters (e.g. mass, length, inertia, mode) , , . The main applications of these results are twofold: the feasibility of an industrial project can be simplified by speeding up the computation of robust controllers and robust margins for systems with rapidly changing architecture parameters, and avoiding usual time-consuming optimization techniques. Secondly, adaptive and embeddable schemes for robust controllers can be proposed and tested while coupling our approach with real-time parameter estimation methods such as the ones developed in the GAIA team. For more details, see .

Preliminary works in the direction have opened
a great variety of questions such as the explicit search for positive
definite solutions of algebraic or differential Riccati equations (i.e. polynomial or differential systems)
with model parameters, the reduction of these
equations, and of the parameters based on symmetries, the
development, of efficient tools for plotting high degree curves and surfaces showing the robustness margins
in terms of the model parameters (collaboration with Moroz (GAMBLE, Inria Nancy)), the use of a certified
numeric Newton-Puiseux algorithm for the design of robust
controllers, etc. ,
, . These results require the use of a
large spectrum of computer algebra methods such as linear algebra
with parameters, polynomial systems with parameters, ordinary
differential systems with parameters, symmetries and reduction,
rational parametrizations, discriminant varieties, semi-algebraic
sets, critical point methods, real root isolation methods, etc. We
shall further develop the parametric robust control in collaboration
with *Safran Electronics & Defense*.

In connection with the above results, parameter estimation methods will be studied to develop *adaptive robust controllers* for
gyrostabilized systems. Indeed, combining explicit characterizations
of robust controllers in terms of the model parameters with
time-to-time estimations of these model parameters (which can change
with the system production, the heat, the wear, etc.), the robust
controllers can then be automatically tuned to conserve their
robustness performances , .

Finally, as explained in , ,
constant and distributed delays naturally appear in *Safran E & D* systems (e.g. gyrostabilized systems using visual trackers, stabilized mirror models). Extensions of the above problems and results will be studied for differential time-delay systems based on robust control techniques for infinite-dimensional systems (see, e.g., and the references therein) and its algebraic extension to include model parameters.

Due to numerous applications (e.g. sensor network, mobile robots),
sources and sensors localization has intensively been studied in the
literature of signal processing. The *anchor position
self calibration problem*, a well-known problem in signal
processing, consists in estimating the positions of both the
moving sources and a set of fixed sensors (anchors) when only the
distance information between the points from the two different sets is
available. The position self-calibration problem is a particular case
of the *Multidimensional Unfolding* (MDU) problem for the
Euclidean space of dimension 3.

Based on computer algebra methods for polynomial
systems, we have recently proposed a new approach for the MDU problem
which yields closed-form solutions and an efficient algorithm for the
estimation of the positions only based on linear
algebra techniques. This first result, obtained in collaboration with
Dagher (Research Engineer, Inria Chile) and Zheng (Defrost,
Inria Lille - Nord Europe), yields a recent *patent* .
Real tests are now carried out. Our first results will be further developed,
improved, tested, and demonstrated.

The MDU problem is just one instance of localization problems: more problems can be addressed
for which a computer algebra expertise can brought new interesting
results, especially in finding closed-form solutions, yielding new
estimation techniques which avoid the use of optimization algorithms as commonly done in the signal processing
literature. The main differences between these localization problems
can essentially be read on a certain matrix of distance called the
*Euclidean distance matrix* .

We have a collaboration with Barau (*Safran Tech*) and Hubert
(*Safran Tech*), and Dagher (Research Engineer, Inria Chile) on
the symbolic-numeric study of the new multi-carrier demodulation
method developed in . *Gear fault diagnosis* is
an important issue in aeronautics industry since a damage in a
gearbox, which is not detected in time, can have dramatic effects on
the safety of a plane.

Since the vibrations of a spur gear can be modeled as a product of two periodic functions related to the gearbox kinematic, has proposed to recover each function from the global signal by means of an optimal reconstruction problem which, by means of Fourier analysis, can be rewritten as

where

For linear systems, the closed-form expressions of the parameters obtained by means of the algebraic parameter estimation problem will continue to provide robust estimates in our multidisciplinary collaborations, in marine biology and human-machine interactions, as it is already the case of existing Non-A results (in collaboration with Loki, Inria Lille - Nord Europe).

For nonlinear systems, a collaboration with biologists and modelers has been developed for a
few years already .
Our partners are a team from the Applied Mathematical Department
of Le Havre University (modelers) and an INSERM team at Rouen
University (neurobiologists). The targeted biological problem is the cortical spreading depression,
a brain disease likely to occur after cerebrovascular accidents .
We seek

**Computer Algebra in Scientific Computing**

The GAIA team organized the conference *Computer Algebra in
Scientific Computing* (CASC), University of Lille, 17–21 September 2018.

*Adaptive Homogeneous Filtering*

Keywords: Automatic differentiation - Filtering

Functional Description: allows to reconstruct a signal based on derivatives estimation and to filter high amplitude and wide frequencies spectrum perturbations.

Contact: Denis Efimov

We have released the first version of the BLINEIDE library. BLINEIDE
stands for Bibliothèques Lilloises dédiées à l'Intégration Numérique
des Équations Intégro-Différentielles. It is an open source C library
dedicated to the numerical integration of systems of
integro-differential equations. for more details, see https://

, present a proof of concept for symbolic and numeric methods dedicated to the parameter estimation problem for models formulated by means of nonlinear integro-differential equations. In particular, we address the computation of the model input-output equation and the numerical integration of integro-differential systems (the BLINEIDE library).

In this joint work with Axel Barrau from Safran Tech , we propose a new step in the development of invariant observers. In the past, this theory led to impressive simplifications of the error equations encountered in estimation problems, especially those related to navigation. This was used to reduce computation load or derive new theoretical properties. Here, we leverage this advantage to obtain closed-form solutions of the underlying algebraic Riccati equations through advanced symbolic computation methods.

This work is made in collaboration with Axel Barrau and Elisa Hubert (Safran Tech), and Roudy Dagher (Research Engineer, Inria Chile). The problem under study, which reduces to a certain signal factorization problem, was shown by Barrau et. al. to be equivalent to a Frobenius norm minimization problem. Starting from this optimization problem, we investigate the use of computer algebra methods to compute explicit solutions for the original problem. Along the way, we exhibit interesting algebraic and geometric properties of the underlying polynomial system. A paper is currently in development to summarize these results.

This work aims to
design a new symbolic-numerical Puiseux-free approach for the study of
the stability of differential time-delay systems. The idea behind is to replace the costly computations of Puiseux developpements around the *critical pairs* of the characteristic function by the numerical analysis of the branches of a well chosen 3D curve. The preliminary results show that this approach is easier to implement and turns out to be more efficient in practice. This ongoing work will be the subject of a future publication.

Within the CIFRE PhD thesis (2014-2018) ,
we have studied new robust stabilization techniques for gyrostabilized systems with unfixed model parameters (e.g. modes, masses,
stiffness of springs, damper magnitudes). Parameters of their models
indeed slowly change with the temperature, fatigue, etc., yielding
time-consuming re-computations of robust controllers. Moreover, the
possibility to quickly know robustness indicators (e.g. margins) and
explicit robust controllers in terms of the model parameters can
highly speed up the design of a project. Finally, closed-form
solutions for robust controllers in terms of the model parameters are
the first steps towards the development of adaptive robust controllers
which can be embedded in gyrostabilized platforms since no
optimization algorithms are then required for a real-time
implementation and only the parameters have to be estimated from time
to time to re-compute the robust controller (based on a basic
arithmetic). To do that, we have introduced algebraic methods and
computer algebra techniques to initiate a new approach entitled
*parametric robust control*. For mor details, see
and
, , . This new approach will be further
developed in the future since it opens both theoretical and
practical interesting questions. In particular, the new PhD thesis of Grace in GAIA aims to study the underlying mathematical problem from both a theoretical and an implementation perspectives.

A new collaboration with Ellcie Healthy, a company based in Nice began in October 2017. It involves the analyze of signals coming from optical sensors installed in glasses. With Denis Efimov, the first studies obtained were very promising. This collaboration was formalized with the signature of a first contract in March 2018. The first objective of this project was to design algorithms for intelligent filtering of data coming from infrared sensors, especially for light-related disturbances. Discussions are currently underway for the submission of new joint projects.

*ANR project MSDOS* (Multidimensional System: Digression on
Stability, coordinator: Nima Yeganefar (Poitiers University),
2014-2018) aimed at studying stability and stabilization problems for
multidimensional systems by means of both analytic and algebraic
methods. For more information, see https://

*ANR TurboTouch* (High-performance touch interactions, coordinator: G. Casiez (MJOLNIR team, Inria), 2014–2019) develops methods and tools on transfer functions to allow high performance tactile interactions (e.g. high precision and low latency) adapted to the user and to the task. This research project is developed in collaboration with the Loki team, Inria Lille – Nord Europe (project leader). For more information, seehttp://

*ANR WaQMoS* (Coastal waters Quality surveillance using bivalve
Mollusk-based Sensors, coordinator: D. Efimov (Non-A Post, Inria), 2015–2020) develops a biosensor, based on measurements and interpretation of bi- valves mollusks behavior, for remote online detection of coastal water pollution and cli- mate change consequences. This research project is developed in collaboration with the Valse team, Inria Lille – Nord Europe (project leader). For more information, see https://

Mohamed Barakat: University of Siegen (Germany)

Effective module theory, effective homological algebra, algebraic analysis, computer algebra, implementation.

Georg Regensburger: Institute for Algebra, Johannes Kepler University Linz (Austria)

Rings of integro-differential-delay operators, computer algebra, implementation.

Daniel Robertz: University of Plymouth (United Kingdom)

Effective algebraic analysis, mathematical systems theory, computer algebra, implementation.

*WeCare*, Inria Northern European Associate Team with the team of
A. Medvedev from Uppsala University on effective algorithms for
estimation and control in wearable devices for health and care,
2018–2020.

We participate in *HoTSMoCE*, an Inria Associated team with Non-A
Post and the team of L. Fridman (UNAM, Mexico), on the development of algebraic and homogeneous tools for sliding mode control and estimation.

Thomas Cluzeau, XLIM, University of Limoges, May 2018.

Marc Moreno Maza, University of Western Ontario, London, Ontario, Canada, September 2018.

Alexander Medvedev, University of Uppsala (03–05/10/2018).

Fredrik Olsson, University of Uppsala (26–30/11/2018).

Elisa Hubert (Safran Tech) visited us twice (23–24/07/2018, 12–13/09/2018) to work on the problem of gear fault diagnostic based on algebraic and symbolic approaches.

The GAIA team organized the 2018 edition of the
*Computer Algebra in Scientific Computing* (CASC) international
workshops series (17–21 September 2018). For more details, see
http://

Since 2018, François Boulier is a *General Chair* of CASC. See http://

A. Quadrat is a member of the *IFAC Technical Committee* “Linear
Control Systems”, International Federation of Automatic Control,
TC2.2.

The GAIA team organized the
*Computer Algebra in Scientific Computing* (CASC), University
of Lille, 17–21 September 2018.

A. Quadrat is a member of the organization committee of the
*Journées Nationales de Calcul Formel* (JNCF), Luminy,
France, 22–26/01/2018.

F. Lemaire was Poster Chair of the *International Symposium
on Symbolic and Algebraic Computation* (ISSAC'2018) Poster
Session, New York, 16–19/05/2018.

Y. Bouzidi reviewed a publication for ISSAC 2018 and for TDS 2018.

F. Lemaire reviewed a publication for CASC 2018.

A. Quadrat reviewed two publications for ISSAC 2018.

A. Quadrat is associate editor of *Multidimensional Systems and
Signal Processing*.

A. Quadrat reviewed papers for the journal *Multidimensional Systems and Signal Processing*.

Y. Bouzidi gave the following talks:

A Symbolic Approach for Solving Algebraic Riccati
Equations, *Journées Nationales de Calcul Formel* (JNCF), CIRM, Marseille - January 22th 2018.

Parametric study of the critical pairs of linear
differential systems with commensurate delays, *Inria DISCO team-project*, Paris - April 30th 2018.

Using symbolic computation to solve algebraic Riccati
equations arising in invariant filtering, *European Control Conference*, Limassol - June 15th 2018.

A symbolic approach for a parametric *visualization group*, ISCD, Paris - Novembre 30th 2018.

A. Quadrat co-organized with N. Yeganefar (University of Poitiers) the invited session
*New Results in Multidimensional Systems Theory* at the
*2018 European Control Conference*, Cyprus, June 12-15, 2018.

R. Ushirobira was co-responsible with Denis Efimov (Non-A Post, Inria) and Gilberto Pin (Electrolux Italia) of a special issue Finite-time estimation, diagnosis and synchronization of uncertain
systems for the *European Journal of Control*.

A. Quadrat is a member of the *Bureau du Comité des
Equipes-Projets* (BCEP) and of the *Commission des Emplois de
Recherche*, Inria Lille - Nord Europe.

A. Quadrat wrote a comparative report for the position “Algorithmische Algebra” (W2), University of Siegen, Germany.

F. Boulier has been Head of the *Spécialité GIS at Polytech Lille* since June 2018.

A. Quadrat is in charge with C. Jamroz of the *RaWeb 2018* for Inria Lille – Nord Europe.

Rosane Ushirobira taught around 90h at Polytech Lille and University of Lille (Linear algebra, analysis and logic).

PhD: Guillaume Rance, “Commande

PhD in progress: Grace Younes, “Calcul de multiplicités de racines de polynoômes et de quasi-polynoômes”, 01 /10/2018, Alban Quadrat & Yacine Bouzidi & Adrien Poteaux.

Master: Ambroise Fleury (Licence 3 ème année training), “Computation of gcd using AVX”, F. Lemaire.

Master: Guillaume Maitrot (Master 2 training), “Improving the BLINEIDE library with OpenMP”, F. Lemaire.

F. Boulier was referee of the PhD thesis of G. Rance, University Paris Orsay, 09/07/2018.

A. Quadrat was a jury member of the CRCN Inria 2018 competition for the Lille – Nord Europe center.

Since December 2103, R. Ushirobira organize the cycle “30 minutes of science”, a rotating monthly seminar for all researchers at Inria LIlle. On average, 40 people participate in this seminar.

In 2017/2018, Rosane Ushirobira was a referent researcher for *the Math en Jean program* at Arthur Rimbaud College (Villeneuve d'Ascq).

Y. Bouzidi, *Symbolic-numeric method for a parametric control problem*, presentation in the *30 minutes of science Inria event*, Lille - January 10th 2018.