Algebraic Multiplicity

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Silviu-iulian Niculescu – 1st expert 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 AlgebraicMultiplicity 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 AlgebraicMultiplicity 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.

Islam Boussaada – 2nd expert 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 AlgebraicMultiplicity 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 AlgebraicMultiplicity 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.

Wang Sheng-hua – 3rd expert on this subject based on the ideXlab platform

• Spectral Analyse of theTransport Operator with a PeriodicBoundary Conditions in Slab Geometry
Journal of Shangrao Normal College, 2020
Co-Authors: Wang Sheng-hua

Abstract:

The objective of this paper is to analyse some spectral properties of transport operator with anisotropic one-energy inhomogeneous slab geometry in periodic boundary condition. On the fundament of ref.~ [3] It is proved the spectrum of the transport operator only consist of finite isolated eigenvalue which have a finite Algebraic Multiplicity in trip, and to prove the existence of the dominant eigenvalue.

• Singular Transport Equations with a Reflecting Boundary Conditions in Slab Geometry
Journal of Nanchang University, 2020
Co-Authors: Wang Sheng-hua

Abstract:

The objective of this paper is to research singular transport equations with anisotropic continuous energy homogeneous in slab geometry of reflecting boundary conditions.It proves that the transport operator generates a strongly continuous semigroup and the weak compactness properties of the second-order remained term of the Dyson-Phillips expansion for the semigroup in space,and to obtain the spectrum of the transport operator only consist of,at most,finitely many isolated eigenvalues which have a finite Algebraic Multiplicity in trip.

• The Spectral Analyse of Transport Operator with a Perfect Reflecting Boundary Condition in Slab Geometry
Journal of Jiangxi Normal University, 2020
Co-Authors: Wang Sheng-hua

Abstract:

The objective of this paper is to research spectral analyse of transport operator with anisotropic continuous energy homogeneous slab geometry in perfect reflecting boundary condition.It proves the transport operator generates a strongly continuous group and the compactness properties of the second-order remained term of the Dyson-Phillips expansion for the C_0 group,and to obtain the spectrum of the transport operator only consist of finite isolated eigenvalue which have a finite Algebraic Multiplicity in trip Γ,and to prove the existence of the dominant eigenvalue.