## Section: Overall Objectives

### A rich interplay between computer algebra and control theory

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 [50] written by their creators, Buchberger, was published in Bose's book [42] 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 [43].

The differential algebra approach to nonlinear control theory [58], [59], [63], [84]
was a major motivation for the effective study of differential algebra [101], [76]
(differential elimination theory, triangular sets, regular chains, etc.)
[45], [46], [72] 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 [73]
and multidimensional systems theory [42], [43]
have recently been profoundly developed due to the so-called behavior
approach [80], [83]
and the module approach [64], [84].
Based on ideas of *algebraic analysis* [75],
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 [79]
and homological algebra [103], had to be iniated
based on functional elimination techniques (i.e. Gröbner or Janet
basis techniques for noncommutative polynomial rings) [7], [8] (see also [91]).
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.