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

Title:
On the Ore extension ring of differential time-varying delay operators
Authors:
Quadrat, Alban, Ushirobira, Rosane
Contributors:
OUtils de Résolution Algébriques pour la Géométrie et ses ApplicatioNs (OURAGAN), Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Centre Inria de Sorbonne Université, Centre Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Finite-time control and estimation for distributed systems (VALSE), Centre Inria de l'Université de Lille, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)
Source:
Accounting for Constraints in Delay Systems. 12(Accounting for Constraints in Delay Systems, Advances in Delays and Dynamics(ADD), volume 12, Springer, pp. 87-107):87-107
Publisher Information:
CCSD; Springer, 2022.
Publication Year:
2022
Collection:
collection:CNRS
collection:INRIA
collection:INRIA-ROCQ
collection:INRIA-LILLE
collection:INSMI
collection:INRIA_TEST
collection:IMJ
collection:TESTALAIN1
collection:CRISTAL
collection:INRIA2
collection:TDS-MACS
collection:UNIV-LILLE
collection:SORBONNE-UNIVERSITE
collection:SORBONNE-UNIV
collection:CRISTAL-VALSE
collection:SU-SCIENCES
collection:UNIV-PARIS
collection:UNIVERSITE-PARIS
collection:UP-SCIENCES
collection:SU-TI
collection:ALLIANCE-SU
collection:SUPRA_MATHS_INFO
Subject Terms:
ACM: G.: Mathematics of Computing, G.1: NUMERICAL ANALYSIS, G.1.7: Ordinary Differential Equations, G.1.9: Integral Equations, G.1.9.0: Delay equations, ACM: I.: Computing Methodologies, I.1: SYMBOLIC AND ALGEBRAIC MANIPULATION, I.1.1: Expressions and Their Representation, I.1.2: Algorithms, I.1.2.0: Algebraic algorithms, [SPI.AUTO]Engineering Sciences [physics], Automatic, [INFO.INFO-SC]Computer Science [cs], Symbolic Computation [cs.SC], [MATH.MATH-RA]Mathematics [math], Rings and Algebras [math.RA], [MATH.MATH-OA]Mathematics [math], Operator Algebras [math.OA], [MATH.MATH-OC]Mathematics [math], Optimization and Control [math.OC]
Original Identifier:
HAL: hal-03908643
Document Type:
Book bookPart<br />Book sections
Language:
English
ISBN:
978-3-030-89014-8
Relation:
info:eu-repo/semantics/altIdentifier/doi/10.1007/978-3-030-89014-8_5
DOI:
10.1007/978-3-030-89014-8_5
Access URL:
https://inria.hal.science/hal-03908643
https://inria.hal.science/hal-03908643v1/document
https://inria.hal.science/hal-03908643v1/file/decod-2018-final.pdf
Rights:
info:eu-repo/semantics/OpenAccess
Accession Number:
edshal.hal.03908643v1
Database:
HAL

Further Information

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.