Algebraic analysis for the Ore extension ring of differential time-varying delay operators

Alban Quadrat (1) , Rosane Ushirobira (1)

1 NON-A - Non-Asymptotic estimation for online systems

As far as we know, there is no algebraic (polynomial) approach for the study of linear differential time-delay systems in the case of a (sufficiently regular) time-varying delay. Based on the concept of skew polynomial rings developed by Ore in the 30s, the purpose of this paper is to construct the ring of differential time-delay operators as an Ore extension and to analyze its properties. Classical algebraic properties of this ring, such as noetherianity, its homological and Krull dimensions and the existence of Gröbner bases, are characterized in terms of the time-varying delay function. In conclusion, the algebraic analysis approach to linear systems theory allows us to study linear differential time-varying delay systems (e.g. existence of autonomous elements, controllability, parametrizability, flatness, behavioral approach) through methods coming from module theory, homological algebra and constructive algebra.

Mots clés: Computer algebra

Domaines: Anneaux et algèbres [math.RA]; Algèbres d'opérateurs [math.OA]; Optimisation et contrôle [math.OC]; Calcul formel [cs.SC]

Alban Quadrat, Rosane Ushirobira. Algebraic analysis for the Ore extension ring of differential time-varying delay operators. 22nd International Symposium on Mathematical Theory of Networks and Systems (MTNS), Jul 2016, Minneapolis, United States. pp.8. ⟨hal-01415256⟩