Algebraic Multiplicity

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Silviu-iulian Niculescu - One of the best experts on this subject based on the ideXlab platform.

  • Characterizing the Codimension of Zero Singularities for Time-Delay Systems
    Acta Applicandae Mathematicae, 2016
    Co-Authors: Islam Boussaada, Silviu-iulian Niculescu
    Abstract:

    The analysis of time-delay systems mainly relies on detecting and understanding the spectral values bifurcations when crossing the imaginary axis. This paper deals with the zero singularity, essentially when the zero spectral value is multiple. The simplest case in such a configuration is characterized by an Algebraic Multiplicity two and a geometric Multiplicity one, known as the Bogdanov-Takens singularity. Moreover, in some cases the codimension of the zero spectral value exceeds the number of the coupled scalar-differential equations. Nevertheless, to the best of the author’s knowledge, the bounds of such a Multiplicity have not been deeply investigated in the literature. It is worth mentioning that the knowledge of such an information is crucial for nonlinear analysis purposes since the dimension of the projected state on the center manifold is none other than the sum of the dimensions of the generalized eigenspaces associated with spectral values with zero real parts. Motivated by a control-oriented problems, this paper provides an answer to this question for time-delay systems, taking into account the parameters’ Algebraic constraints that may occur in applications. We emphasize the link between such a problem and the incidence matrices associated with the Birkhoff interpolation problem. In this context, symbolic algorithms for LU-factorization for functional confluent Vandermonde as well as some classes of bivariate functional Birkhoff matrices are also proposed.

  • tracking the Algebraic Multiplicity of crossing imaginary roots for generic quasipolynomials a vandermonde based approach
    IEEE Transactions on Automatic Control, 2016
    Co-Authors: Islam Boussaada, Silviu-iulian Niculescu
    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 $j\omega$ where $\omega\neq 0$ 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-Szego generic bound arising from the principle argument is established.

  • Tracking the Algebraic Multiplicity of Crossing Imaginary Roots for Generic Quasipolynomials: A Vandermonde-Based Approach
    IEEE Transactions on Automatic Control, 2016
    Co-Authors: Islam Boussaada, Silviu-iulian Niculescu
    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.

  • Computing the codimension of the singularity at the origin for delay systems in the regular case: A vandermonde-based approach
    2014 European Control Conference (ECC), 2014
    Co-Authors: Islam Boussaada, Dina-alina Irofti, Silviu-iulian Niculescu
    Abstract:

    The analysis of time-delay systems mainly relies on the identification and the understanding of the spectral values bifurcations when crossing the imaginary axis. One of the most important type of such singularities is when the zero spectral value is multiple. The simplest case in such a configuration is characterized by an Algebraic Multiplicity two and a geometric Multiplicity one known as Bogdanov-Takens singularity. Moreover, in some circumstances the codimension of the zero spectral value exceeds the dimension of the delay-free system of differential equations. To the best of the authors' knowledge, the bound of such a Multiplicity was not deeply investigated in the literature. This paper provides an answer to this question for time-delay systems with linear part characterized in the Laplace domain by a quasipolynomial function with non sparse polynomials and without coupling delays.

Islam Boussaada - One of the best experts on this subject based on the ideXlab platform.

  • Characterizing the Codimension of Zero Singularities for Time-Delay Systems
    Acta Applicandae Mathematicae, 2016
    Co-Authors: Islam Boussaada, Silviu-iulian Niculescu
    Abstract:

    The analysis of time-delay systems mainly relies on detecting and understanding the spectral values bifurcations when crossing the imaginary axis. This paper deals with the zero singularity, essentially when the zero spectral value is multiple. The simplest case in such a configuration is characterized by an Algebraic Multiplicity two and a geometric Multiplicity one, known as the Bogdanov-Takens singularity. Moreover, in some cases the codimension of the zero spectral value exceeds the number of the coupled scalar-differential equations. Nevertheless, to the best of the author’s knowledge, the bounds of such a Multiplicity have not been deeply investigated in the literature. It is worth mentioning that the knowledge of such an information is crucial for nonlinear analysis purposes since the dimension of the projected state on the center manifold is none other than the sum of the dimensions of the generalized eigenspaces associated with spectral values with zero real parts. Motivated by a control-oriented problems, this paper provides an answer to this question for time-delay systems, taking into account the parameters’ Algebraic constraints that may occur in applications. We emphasize the link between such a problem and the incidence matrices associated with the Birkhoff interpolation problem. In this context, symbolic algorithms for LU-factorization for functional confluent Vandermonde as well as some classes of bivariate functional Birkhoff matrices are also proposed.

  • tracking the Algebraic Multiplicity of crossing imaginary roots for generic quasipolynomials a vandermonde based approach
    IEEE Transactions on Automatic Control, 2016
    Co-Authors: Islam Boussaada, Silviu-iulian Niculescu
    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 $j\omega$ where $\omega\neq 0$ 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-Szego generic bound arising from the principle argument is established.

  • Tracking the Algebraic Multiplicity of Crossing Imaginary Roots for Generic Quasipolynomials: A Vandermonde-Based Approach
    IEEE Transactions on Automatic Control, 2016
    Co-Authors: Islam Boussaada, Silviu-iulian Niculescu
    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.

  • Computing the codimension of the singularity at the origin for delay systems in the regular case: A vandermonde-based approach
    2014 European Control Conference (ECC), 2014
    Co-Authors: Islam Boussaada, Dina-alina Irofti, Silviu-iulian Niculescu
    Abstract:

    The analysis of time-delay systems mainly relies on the identification and the understanding of the spectral values bifurcations when crossing the imaginary axis. One of the most important type of such singularities is when the zero spectral value is multiple. The simplest case in such a configuration is characterized by an Algebraic Multiplicity two and a geometric Multiplicity one known as Bogdanov-Takens singularity. Moreover, in some circumstances the codimension of the zero spectral value exceeds the dimension of the delay-free system of differential equations. To the best of the authors' knowledge, the bound of such a Multiplicity was not deeply investigated in the literature. This paper provides an answer to this question for time-delay systems with linear part characterized in the Laplace domain by a quasipolynomial function with non sparse polynomials and without coupling delays.

Wang Sheng-hua - One of the best experts on this subject based on the ideXlab platform.

Guo-dong Zhang - One of the best experts on this subject based on the ideXlab platform.

  • On the spectrum of Euler–Bernoulli beam equation with Kelvin–Voigt damping☆
    Journal of Mathematical Analysis and Applications, 2011
    Co-Authors: Guo-dong Zhang
    Abstract:

    Abstract The spectral property of an Euler–Bernoulli beam equation with clamped boundary conditions and internal Kelvin–Voigt damping is considered. The essential spectrum of the system operator is rigorously identified to be an interval on the left real axis. Under some assumptions on the coefficients, it is shown that the essential spectrum contains continuous spectrum only, and the point spectrum consists of isolated eigenvalues of finite Algebraic Multiplicity. The asymptotic behavior of eigenvalues is presented.

  • Spectral analysis of a wave equation with Kelvin-Voigt damping
    Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik, 2010
    Co-Authors: Jun-min Wang, Guo-dong Zhang
    Abstract:

    A vibrating system with some kind of internal damping represents a distributed or passive control. In this article, a wave equation with clamped boundary conditions and internal Kelvin-Voigt damping is considered. It is shown that the spectrum of the system operator is composed of two parts: point spectrum and continuous spectrum. The point spectrum consists of isolated eigenvalues of finite Algebraic Multiplicity, and the continuous spectrum that is identical to the essential spectrum is an interval on the left real axis. The asymptotic behavior of eigenvalues is presented.

  • CDC - Frequency analysis of a wave equation with Kelvin-Voigt damping
    Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference, 2009
    Co-Authors: Jun-min Wang, Guo-dong Zhang
    Abstract:

    A vibrating system with some kind of internal damping represents a distributed or passive control. In this article, a wave equation with clamped boundary conditions and internal Kelvin-Voigt damping is considered. It is shown that the spectrum of system operator is composed of two parts: point spectrum and continuous spectrum. The point spectrum is consist of isolated eigenvalues of finite Algebraic Multiplicity, and the continuous spectrum is an interval on the left real axis. The asymptotic behavior of eigenvalues is presented.

  • Frequency analysis of a wave equation with Kelvin-Voigt damping
    Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference, 2009
    Co-Authors: Jun-min Wang, Guo-dong Zhang
    Abstract:

    A vibrating system with some kind of internal damping represents a distributed or passive control. In this article, a wave equation with clamped boundary conditions and internal Kelvin-Voigt damping is considered. It is shown that the spectrum of system operator is composed of two parts: point spectrum and continuous spectrum. The point spectrum is consist of isolated eigenvalues of finite Algebraic Multiplicity, and the continuous spectrum is an interval on the left real axis. The asymptotic behavior of eigenvalues is presented.

Wu Hong-xing - One of the best experts on this subject based on the ideXlab platform.