On the computation of π\pi-flat outputs for differential-delay systems

We introduce a new definition of $\pi$-flatness for linear differential delay systems with time-varying coefficients. We characterize $\pi$- and $\pi$-0-flat outputs and provide an algorithm to efficiently compute such outputs. We present an academic example of motion planning to discuss the pertinence of the approach.Comment: Minor corrections to fit with the journal versio

Une approche par l’analyse algébrique effectivedes systèmes linéaires sur des algèbres de Ore

The purpose of this paper is to present a survey on the effective algebraic analysis approach to linear systems theory with applications to control theory and mathematical physics. In particular, we show how the combination of effective methods of computer algebra - based on Gröbner basis techniques over a class of noncommutative polynomial rings of functional operators called Ore algebras - and constructive aspects of module theory and homological algebra enables the characterization of structural properties of linear functional systems. Algorithms are given and a dedicated implementation, called OreAlgebraicAnalysis, based on the Mathematica package HolonomicFunctions, is demonstrated.Le but de ce papier est de présenter un état de l’art d’une approche par l’analyse algébrique effective de la théorie des systèmes linéaires avec des applications à la théorie du contrôle et à la physique mathématique.En particulier, nous montrons comment la combinaison des méthodes effectives de calcul formel - basées sur lestechniques de bases de Gröbner sur une classe d’algèbres polynomiales noncommutatives d’opérateurs fonctionnels appelée algèbres de Ore - et d’aspects constructifs de théorie des modules et d’algèbre homologique permet lacaractérisation de propriétés structurelles des systèmes linéaires fonctionnels. Des algorithmes sont donnés et uneimplémentation dédiée, appelée OREALGEBRAICANALYSIS, basée sur le package Mathematica HOLONOMIC-FUNCTIONS, est présenté

Signature Gr\"obner bases in free algebras over rings

We generalize signature Gr\"obner bases, previously studied in the free algebra over a field or polynomial rings over a ring, to ideals in the mixed algebra $R[x_1,...,x_k]\langle y_1,\dots,y_n \rangle$ where $R$ is a principal ideal domain. We give an algorithm for computing them, combining elements from the theory of commutative and noncommutative (signature) Gr\"obner bases, and prove its correctness. Applications include extensions of the free algebra with commutative variables, e.g., for homogenization purposes or for performing ideal theoretic operations such as intersections, and computations over $\mathbb{Z}$ as universal proofs over fields of arbitrary characteristic. By extending the signature cover criterion to our setting, our algorithm also lifts some technical restrictions from previous noncommutative signature-based algorithms, now allowing, e.g., elimination orderings. We provide a prototype implementation for the case when $R$ is a field, and show that our algorithm for the mixed algebra is more efficient than classical approaches using existing algorithms.Comment: 10 page