Tracking the Algebraic Multiplicity of Crossing Imaginary Roots for Generic Quasipolynomials: A Vandermonde-Based Approach

Islam Boussaada 1, 2, 3 Silviu-Iulian Niculescu 1, 2
2 DISCO - Dynamical Interconnected Systems in COmplex Environments
L2S - Laboratoire des signaux et systèmes, Inria Saclay - Ile de France, SUPELEC, CNRS - Centre National de la Recherche Scientifique : UMR8506
Abstract : A standard approach in analyzing dynamical systems consists in identifying and understanding the eigenvalues bifurcations when crossing the imaginary axis. Efficient methods for crossing imaginary roots identification exist. However, to the best of the author's knowledge, the multiplicity of such roots was not deeply investigated. In recent papers by the authors [1], [2], it is emphasized that the multiplicity of the zero spectral value can exceed the number of the coupled scalar delay-differential equations and a constructive approach Vandermonde-based allowing to an adaptive bound for such a multiplicity is provided. Namely, it is shown that the zero spectral value multiplicity depends on the system structure (number of delays and number of non zero coefficients of the associated quasipolynomial) rather than the degree of the associated quasipolynomial [3]. This technical note extends the constructive approach in investigating the multiplicity of crossing imaginary roots jω where ω ≠ 0 and establishes a link with a new class of functional confluent Vandermonde matrices. A symbolic algorithm for computing the LU-factorization for such matrices is provided. As a byproduct of the proposed approach, a bound sharper than the Polya-Szegö generic bound arising from the principle argument is established.
Type de document :
Article dans une revue
IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, 2016, 61 (6), pp.1601-1606. 〈10.1109/TAC.2015.2480175〉
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https://hal-centralesupelec.archives-ouvertes.fr/hal-01425757
Contributeur : Islam Boussaada <>
Soumis le : mardi 3 janvier 2017 - 18:59:31
Dernière modification le : jeudi 5 avril 2018 - 12:30:14

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Islam Boussaada, Silviu-Iulian Niculescu. Tracking the Algebraic Multiplicity of Crossing Imaginary Roots for Generic Quasipolynomials: A Vandermonde-Based Approach. IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, 2016, 61 (6), pp.1601-1606. 〈10.1109/TAC.2015.2480175〉. 〈hal-01425757〉

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