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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.

  • computing the codimension of the singularity at the origin for delay systems in the regular case a vandermonde based approach
    European Control Conference, 2014
    Co-Authors: Islam Boussaada, Dinaalina Irofti, Silviu-iulian Niculescu
    Abstract:

    A standard framework in analyzing Time-delay systems consists first, in identifying the associated crossing roots and secondly, then, in characterizing the local bifurcations of such roots with respect to small variations of the system parameters. Moreover, the dynamics of such spectral values are strongly related to their multiplicities (algebraic/Geometric). This paper focuses on an interesting type of such singularities; that is when the zero spectral value is multiple. The simplest case,whichisquitecommoninapplications,ischaracterizedby an algebraic Multiplicity two and a Geometric Multiplicity one known as Bogdanov-Takens singularity. Unlike finite dimen- sional systems, the algebraic Multiplicity of the zero spectral value may exceed the dimension of the delay-free system of differential equations. To the best of the authors’ knowledge, the bound of such a Multiplicity for Time-delay systems was not deeply investigated in the literature. Our contribution is two fold. First, we emphasize the link between the Multiplicity characterization and Birkhoff matrices. Secondly, we elaborate a constructive bound for the zero spectral value in the regular case; i.e. when the delay polynomials of a given quasipolynomial are complete, as well as in the singular case; i.e. when such polynomials are sparse. In the last case, the established bound is sharper than Polya-Szego generic bound.

Yong Chen - One of the best experts on this subject based on the ideXlab platform.

  • jordan decomposition and Geometric Multiplicity for a class of non symmetric ornstein uhlenbeck operators
    Advances in Difference Equations, 2014
    Co-Authors: Jiying Wang, Yong Chen
    Abstract:

    In this paper, we calculate the Jordan decomposition for a class of non-symmetric Ornstein-Uhlenbeck operators with the drift coefficient matrix, being a Jordan block, and the diffusion coefficient matrix, being the identity multiplying a constant. For the 2-dimensional case, we present all the general eigenfunctions by mathematical induction. For the 3-dimensional case, we divide the calculation of the Jordan decomposition into three steps. The key step is to do the canonical projection onto the homogeneous Hermite polynomials, and then use the theory of systems of linear equations. Finally, we get the Geometric Multiplicity of the eigenvalue of the Ornstein-Uhlenbeck operator.

  • on the jordan decomposition for a class of non symmetric ornstein uhlenbeck operators
    arXiv: Probability, 2012
    Co-Authors: Yong Chen, Ying Li
    Abstract:

    In this paper, we calculate the Jordan decomposition (or say, the Jordan canonical form) for a class of non-symmetric Ornstein-Uhlenbeck operators with the drift coefficient matrix being a Jordan block and the diffusion coefficient matrix being identity multiplying a constant. For the 2-dimensional case, we present all the general eigenfunctions by the induction. For the 3-dimensional case, we divide the calculating of the Jordan decomposition into several steps (the key step is to do the canonical projection onto the homogeneous Hermite polynomials, next we use the theory of systems of linear equations). As a by-pass product, we get the Geometric Multiplicity of the eigenvalue of the Ornstein-Uhlenbeck operator.

Dette Holger - One of the best experts on this subject based on the ideXlab platform.

  • Adaptive grid semidefinite programming for finding optimal designs
    eScholarship University of California, 2017
    Co-Authors: Duarte, Belmiro P.m., Wong, Weng Kee, Dette Holger
    Abstract:

    We find optimal designs for linear models using anovel algorithm that iteratively combines a semidefinite programming(SDP) approach with adaptive grid techniques.The proposed algorithm is also adapted to find locally optimaldesigns for nonlinear models. The search space is firstdiscretized, and SDP is applied to find the optimal designbased on the initial grid. The points in the next grid set arepoints that maximize the dispersion function of the SDPgeneratedoptimal design using nonlinear programming. Theprocedure is repeated until a user-specified stopping rule isreached. The proposed algorithm is broadly applicable, andwe demonstrate its flexibility using (i) models with one ormore variables and (ii) differentiable design criteria, suchas A-, D-optimality, and non-differentiable criterion like Eoptimality,including the mathematically more challengingcasewhen theminimum eigenvalue of the informationmatrixof the optimal design has Geometric Multiplicity larger than 1. Our algorithm is computationally efficient because it isbased on mathematical programming tools and so optimalityis assured at each stage; it also exploits the convexity of theproblems whenever possible. Using several linear and nonlinearmodelswith one or more factors, we showthe proposedalgorithm can efficiently find optimal designs

  • Adaptive grid semidefinite programming for finding optimal designs
    2016
    Co-Authors: Duarte, Belmiro P.m., Wong, Weng Kee, Dette Holger
    Abstract:

    We find optimal designs for linear models using a novel algorithm that iteratively combines a Semidefinite Programming (SDP) approach with adaptive grid (AG) techniques. The search space is first discretized and SDP is applied to find the optimal design based on the initial grid. The points in the next grid set are points that maximize the dispersion function of the SDP-generated optimal design using Nonlinear Programming (NLP). The procedure is repeated until a user-specified stopping rule is reached. The proposed algorithm is broadly applicable and we demonstrate its flexibility using (i) models with one or more variables, and (ii) differentiable design criteria, such as A-, D-optimality, and non-differentiable criterion like E-optimality, including the mathematically more challenging case when the minimum eigenvalue of the information matrix of the optimal design has Geometric Multiplicity larger than 1. Our algorithm is computationally efficient because it is based on mathematical programming tools and so optimality is assured at each stage; it also exploits the convexity of the problems whenever possible. Using several linear models, we show the proposed algorithm can efficiently find both old and new optimal designs

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.

  • computing the codimension of the singularity at the origin for delay systems in the regular case a vandermonde based approach
    European Control Conference, 2014
    Co-Authors: Islam Boussaada, Dinaalina Irofti, Silviu-iulian Niculescu
    Abstract:

    A standard framework in analyzing Time-delay systems consists first, in identifying the associated crossing roots and secondly, then, in characterizing the local bifurcations of such roots with respect to small variations of the system parameters. Moreover, the dynamics of such spectral values are strongly related to their multiplicities (algebraic/Geometric). This paper focuses on an interesting type of such singularities; that is when the zero spectral value is multiple. The simplest case,whichisquitecommoninapplications,ischaracterizedby an algebraic Multiplicity two and a Geometric Multiplicity one known as Bogdanov-Takens singularity. Unlike finite dimen- sional systems, the algebraic Multiplicity of the zero spectral value may exceed the dimension of the delay-free system of differential equations. To the best of the authors’ knowledge, the bound of such a Multiplicity for Time-delay systems was not deeply investigated in the literature. Our contribution is two fold. First, we emphasize the link between the Multiplicity characterization and Birkhoff matrices. Secondly, we elaborate a constructive bound for the zero spectral value in the regular case; i.e. when the delay polynomials of a given quasipolynomial are complete, as well as in the singular case; i.e. when such polynomials are sparse. In the last case, the established bound is sharper than Polya-Szego generic bound.

Duarte, Belmiro P.m. - One of the best experts on this subject based on the ideXlab platform.

  • Adaptive grid semidefinite programming for finding optimal designs
    eScholarship University of California, 2017
    Co-Authors: Duarte, Belmiro P.m., Wong, Weng Kee, Dette Holger
    Abstract:

    We find optimal designs for linear models using anovel algorithm that iteratively combines a semidefinite programming(SDP) approach with adaptive grid techniques.The proposed algorithm is also adapted to find locally optimaldesigns for nonlinear models. The search space is firstdiscretized, and SDP is applied to find the optimal designbased on the initial grid. The points in the next grid set arepoints that maximize the dispersion function of the SDPgeneratedoptimal design using nonlinear programming. Theprocedure is repeated until a user-specified stopping rule isreached. The proposed algorithm is broadly applicable, andwe demonstrate its flexibility using (i) models with one ormore variables and (ii) differentiable design criteria, suchas A-, D-optimality, and non-differentiable criterion like Eoptimality,including the mathematically more challengingcasewhen theminimum eigenvalue of the informationmatrixof the optimal design has Geometric Multiplicity larger than 1. Our algorithm is computationally efficient because it isbased on mathematical programming tools and so optimalityis assured at each stage; it also exploits the convexity of theproblems whenever possible. Using several linear and nonlinearmodelswith one or more factors, we showthe proposedalgorithm can efficiently find optimal designs

  • Adaptive grid semidefinite programming for finding optimal designs
    2016
    Co-Authors: Duarte, Belmiro P.m., Wong, Weng Kee, Dette Holger
    Abstract:

    We find optimal designs for linear models using a novel algorithm that iteratively combines a Semidefinite Programming (SDP) approach with adaptive grid (AG) techniques. The search space is first discretized and SDP is applied to find the optimal design based on the initial grid. The points in the next grid set are points that maximize the dispersion function of the SDP-generated optimal design using Nonlinear Programming (NLP). The procedure is repeated until a user-specified stopping rule is reached. The proposed algorithm is broadly applicable and we demonstrate its flexibility using (i) models with one or more variables, and (ii) differentiable design criteria, such as A-, D-optimality, and non-differentiable criterion like E-optimality, including the mathematically more challenging case when the minimum eigenvalue of the information matrix of the optimal design has Geometric Multiplicity larger than 1. Our algorithm is computationally efficient because it is based on mathematical programming tools and so optimality is assured at each stage; it also exploits the convexity of the problems whenever possible. Using several linear models, we show the proposed algorithm can efficiently find both old and new optimal designs