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GAIA Research team

Geometry, Algebra, Informatics, Applications

Team presentation

Functional systems are systems whose unknowns are functions, such as systems of ordinary or partial differential equations, of differential time-delay equations, of difference equations, of integro-differential equations. These systems play a fundamental role in the mathematical modeling of physical phenomena studied in mathematical physics, engineering sciences, mathematical biology, etc. Numerical simulations are usually based on mathematical models defined by functional equations. Functional systems are also the cornerstone of many domains, for instance in mathematical physics, mathematical systems theory, control theory and signal processing.

Numerical aspects of functional systems, especially differential systems, have been widely studied in applied mathematics due to numerical simulation issues. Complementary approaches, based on algebraic and differential geometric methods, are usually upstream or help the numerical simulation. These methods tackle a different range of questions and problems such as algebraic preconditioning, elimination and simplification, completion to formal integrability or involution, computation of integrability conditions and compatibility conditions, index reduction, reduction of variables, choice of adapted coordinate systems based on symmetries, computation of first integrals of motion and conservation laws, study of the (asymptotic) behaviour of solutions at a singularity, or of their dependency with respect to system parameters. Although not yet very popular in applied mathematics, algebraic and differential geometric methods have lengthly been studied in pure mathematics and, over the past years, in computer algebra within an effective viewpoint, mostly driven by applications.

Differential elimination techniques based on differential algebra or Spencer's theory for differential systems, or Gröbner or Janet basis methods for noncommutative polynomial rings of functional operators for linear functional systems are remarkable examples of these algebraic or geometric techniques. They form the algorithmic "engines"' at the basis of recent effective versions of algebraic theories developed in mathematics. In the computer algebra community, a major source of motivation for the development of the effective study of these theories is represented by control theory issues. Certain problems studied in control theory can indeed be better understood and finely studied by means of algebraic or geometric structures, and by means of algebraic or differential geometric techniques. To effectively study the original problems, computer algebra methods allow one to design efficient algorithms that can be implemented in symbolic and symbolic-numeric software.

Research themes

The principal goal of the GAIA team is to study the systems defined by functional equations (i.e., systems whose unknowns are functions), particularly systems of ordinary differential equations, systems of differential time-delay equations and systems of integro-differential equations, by means of algebraic methods, computer algebra (symbolic and symbolic-numeric methods) and mathematical systems theory. The systems to be investigated can be linear, nonlinear, continuous, discrete, or originated from real life applications.

The second goal is to study important problems coming from

  • control theory (e.g. parametric robust control, stabilization of multidimensional systems and differential time-delay systems)
  • signal processing (e.g. parameter estimation problems, metric multidimensional unfolding)
  • multidisciplinary domains (e.g. marine bivalves behaviour, human-machine interaction, ionic activities in neuroscience)

in which functional systems play a fundamental role.

The third goal is to develop dedicated packages for functional systems and their applications and in parallel, eventual industry transfer.

International and industrial relations

ANR France-Germany PRCI Symbiont (Symbolic Methods for Biological Networks):

ANR MSDOS (Multidimensional Systems: Digression On Stability):

ANR WaQMoS (Coastal waters quality surveillance using bivalve mollusk-based sensors):

ANR TurboTouch: High-performance touch interactions:

SAGEM Défense Sécurité, Massy-Palaiseau, Stabilisation paramétrique des systèmes flexibles à retard et applications aux viseurs, CIFRE PhD thesis

PHC Amadeus Computer Algebra for linear Functional Equations:

Collaborations with Linz (Austria), Tampere (Finland), RWTH Aachen (Germany), Bonn (Germany), Kassel (Germany), Siegen (Germany), UNAM (Mexico), Plymouth (U.K.), Ouragan (Inria Paris), MAMBA (Inria Paris)

Keywords: Algebraic methods Computer algebra Functional systems Systems of integro-differential equations Stabilisation Parameter estimation Applications to multidisciplinary domains