On the Ore extension ring of differential time-varying delay operators

Alban Quadrat (1, 2) , Rosane Ushirobira (3)

1 OURAGAN - OUtils de Résolution Algébriques pour la Géométrie et ses ApplicatioNs
2 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
3 VALSE - Finite-time control and estimation for distributed systems

In this work, we propose an algebraic method to study linear differential time-varying delay (DTVD) systems. Our goal is to give an effective construction of the ring of DTVD operators as an Ore extension, thanks to the concept of skew polynomial rings developed by Ore in the 30s. Some algebraic properties of the DTVD operators ring are analyzed, such as its Noetherianity, its homological and Krull dimensions, and the existence of Gröbner bases, all given in terms of the time-varying delay function. The algebraic analysis framework for linear systems theory allows us to study linear DTVD systems and essential properties such as the existence of autonomous elements, controllability, parametrizability, flatness, etc., through methods coming from module theory, homological algebra, and constructive algebra.

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

Alban Quadrat, Rosane Ushirobira. On the Ore extension ring of differential time-varying delay operators. Accounting for Constraints in Delay Systems, Advances in Delays and Dynamics (ADD), volume 12, Springer, pp. 87-107, ADD-2, Springer, pp. 87-107, 2022, Advances in Delays and Dynamics, 978-3-030-89014-8. ⟨10.1007/978-3-030-89014-8_5⟩. ⟨hal-03908643⟩