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Maria Korotyaeva - One of the best experts on this subject based on the ideXlab platform.

  • Revealing the dispersion of phonons from stochastic excitation of wave propagation
    2018
    Co-Authors: Vincent Laude, Maria Korotyaeva
    Abstract:

    The band structure is a central concept in the field of phononics. Indeed, capturing the dispersion of Bloch waves gives invaluable information on allowed propagation modes, their velocity, the existence of local resonances, and the occurrence of band gaps. A band structure are usually obtained by solving an eigenvalue problem that is defined on a closed and bounded domain, resulting in a discrete spectrum. As the wavenumber is varied continuously, the eigenvalues form bands in the dispersion relation. There are at least two cases, however, that resist reduction to a linear eigenvalue problem: first, when dispersive material loss is taken into account and second, when the unit-cell of the crystal extends beyond any bound, as in the case of phononic crystal of holes or pillars on a semi-infinite substrate. We introduce a technique to obtain the phononic band structure that does not rely on searching for eigenvalues, but instead produces a mapping of the Resolvent Set in dispersion space. In spectral theory, indeed, the spectrum is the singular complement of the Resolvent Set in the complex plane. The idea is then to obtain the Resolvent Set of the dynamical Helmholtz equation as a function of wavenumber. The method has been implemented with finite element analysis and has been applied to several problems in phononic crystal theory. In the case of dispersive loss, the complex poles of the density of states give a direct account of propagation loss of each dispersion branch as a function of frequency and wavenumber. In the case of phononic crystals of finite-depth holes or of finite-height pillars sitting on a semi-infinite substrate, the dispersion inside the sound cone - or radiative region - is obtained and leaky guided waves can be identified.

  • Stochastic excitation method for calculating the Resolvent band structure of periodic media and waveguides
    Physical Review B: Condensed Matter and Materials Physics, 2018
    Co-Authors: Vincent Laude, Maria Korotyaeva
    Abstract:

    We introduce a stochastic excitation method for calculating the dispersion relation for waves propagating in periodic media or along waveguides and subject to material loss or radiation damping. Instead of looking for an explicit or implicit functional relation between frequency ω and wave number k, as is usually done, we consider a mapping of the Resolvent Set in the dispersion space (ω,k). Bands appear as the trace of Lorentzian responses containing local information on propagation loss in both time and space domains. For illustration purposes, the method is applied to a lossy sonic crystal, a radiating surface phononic crystal, and a radiating optical waveguide. The Resolvent band structure can be obtained for any system described by a time-harmonic wave equation.

Ruth F. Curtain - One of the best experts on this subject based on the ideXlab platform.

  • Classes of Semigroups
    Introduction to Infinite-Dimensional Systems Theory, 2020
    Co-Authors: Ruth F. Curtain, Hans Zwart
    Abstract:

    The general concept of strongly continuous semigroup was introduced in the previous chapter. In this chapter we focus on examples. This is done by studying three main classes, namely spatially invariant operators, Riesz-spectral operators, and delay equations. Next to existence of the strongly continuous semigroup, we study for these classes the Resolvent Set and operator, and characterise the semigroup invariant subspaces. The chapter ends with a Set of 25 exercises and a notes and references section.

  • Spectral properties of pseudo-Resolvents under structured perturbations
    Mathematics of Control Signals and Systems, 2009
    Co-Authors: Ruth F. Curtain, Birgit Jacob
    Abstract:

    The changes in the spectrum caused by structured perturbations of pseudo-Resolvents and operators on Banach spaces are considered. In particular, if a point is in the Resolvent Set of an operator, necessary and sufficient conditions for it to remain in the Resolvent Set under structured perturbations are given. The structured perturbations of an operator are specified by an operator node that has three generating operators and a characteristic function together with an admissible feedback operator. In addition, the robustness of stability under structured perturbations is analyzed. The results are applied to boundary control systems and impedance passive systems.

  • Regular linear systems and their reciprocals: applications to Riccati equations
    Systems & Control Letters, 2003
    Co-Authors: Ruth F. Curtain
    Abstract:

    For a regular linear system with zero in the Resolvent Set of the generator we introduce its reciprocal system, which has bounded generators. We show that there are close relationships between key system theoretic properties of such regular linear systems and their reciprocals. To illustrate the usefulness of this connection we give conditions under which their respective Riccati equations have the same self-adjoint bounded solutions.

  • Riccati Equations for Stable Well-Posed Linear Systems: The Generic Case
    SIAM Journal on Control and Optimization, 2003
    Co-Authors: Ruth F. Curtain
    Abstract:

    Under the generic assumption that zero is in the Resolvent Set of the generator, we show that the optimal control problem for a stable well-posed linear system is equivalent to a control problem for its reciprocal system which has bounded generating operators. Consequently, the operator X that defines the optimal cost satisfies a Riccati equation with bounded operators. Previous results needed various regularity assumptions to obtain X as a solution to a Riccati equation resembling that in the finite-dimensional theory.

Geni Gupur - One of the best experts on this subject based on the ideXlab platform.

  • Dynamic Analysis of the M/G/1 Queueing Model with Single Working Vacation
    International Journal of Applied and Computational Mathematics, 2016
    Co-Authors: Ehmet Kasim, Geni Gupur
    Abstract:

    By using the strong continuous semigroup theory of linear operators we prove that the M/G/1 queueing model with single working vocation has a unique nonnegative time-dependent solution. When the service completion rate are constant, by studying spectral properties of the operator corresponding to the model we study asymptotic behavior of its time-dependent solution. Fist of all, through studying the Resolvent Set of the adjoint operator of the operator we obtain that all points on the imaginary axis except zero belong to the Resolvent Set of the operator. Next, we prove that zero is eigenvalue of the underlying operator and its adjoint operator. Therefore, by combining these results we deduce that the time-dependent solution of the model strongly converges to its steady-state solution.

  • Further Research on the M/G/1 Retrial Queueing Model with Server Breakdowns
    Journal of Applied Mathematics, 2012
    Co-Authors: Ehmet Kasim, Geni Gupur
    Abstract:

    We study spectral properties of the operator which corresponds to the M/G/1 retrial queueing model with server breakdowns and obtain that all points on the imaginary axis except zero belong to the Resolvent Set of the operator and 0 is not an eigenvalue of the operator. Our results show that the time-dependent solution of the model is probably strongly asymptotically stable.

  • Analysis of the M/G/1 retrial queueing model with server breakdowns
    Journal of Pseudo-Differential Operators and Applications, 2010
    Co-Authors: Geni Gupur
    Abstract:

    By using the Hille–Yosida theorem and the Phillips theorem in functional analysis we prove that the M/G/1 retrial queueing model with server breakdowns has a unique nonnegative time-dependent solution. Next, when the service completion rate is a constant, by studying spectral properties of the operator corresponding to the model we study asymptotic behavior of its time-dependent solution. First of all, through considering the Resolvent Set of the adjoint operator of the operator we obtain that all points on the imaginary axis except zero belong to the Resolvent Set of the operator. In addition, we prove that zero is not an eigenvalue of the operator. Our results show that the time-dependent solution of the model strongly converges to zero.

  • Resolvent Set of the M/MB/1 operator
    Computers & Mathematics with Applications, 2002
    Co-Authors: Geni Gupur
    Abstract:

    Abstract First, we prove that the M/MB/1 operator is a conservative operator, thus showing that the M/MB/1 queueing model has a unique positive time-dependent solution which satisfies the probability condition, and then prove that all points on the imaginary axis except for zero belong to the Resolvent Set of this operator.

Peter Yuditskii - One of the best experts on this subject based on the ideXlab platform.

Vincent Laude - One of the best experts on this subject based on the ideXlab platform.

  • Revealing the dispersion of phonons from stochastic excitation of wave propagation
    2018
    Co-Authors: Vincent Laude, Maria Korotyaeva
    Abstract:

    The band structure is a central concept in the field of phononics. Indeed, capturing the dispersion of Bloch waves gives invaluable information on allowed propagation modes, their velocity, the existence of local resonances, and the occurrence of band gaps. A band structure are usually obtained by solving an eigenvalue problem that is defined on a closed and bounded domain, resulting in a discrete spectrum. As the wavenumber is varied continuously, the eigenvalues form bands in the dispersion relation. There are at least two cases, however, that resist reduction to a linear eigenvalue problem: first, when dispersive material loss is taken into account and second, when the unit-cell of the crystal extends beyond any bound, as in the case of phononic crystal of holes or pillars on a semi-infinite substrate. We introduce a technique to obtain the phononic band structure that does not rely on searching for eigenvalues, but instead produces a mapping of the Resolvent Set in dispersion space. In spectral theory, indeed, the spectrum is the singular complement of the Resolvent Set in the complex plane. The idea is then to obtain the Resolvent Set of the dynamical Helmholtz equation as a function of wavenumber. The method has been implemented with finite element analysis and has been applied to several problems in phononic crystal theory. In the case of dispersive loss, the complex poles of the density of states give a direct account of propagation loss of each dispersion branch as a function of frequency and wavenumber. In the case of phononic crystals of finite-depth holes or of finite-height pillars sitting on a semi-infinite substrate, the dispersion inside the sound cone - or radiative region - is obtained and leaky guided waves can be identified.

  • Stochastic excitation method for calculating the Resolvent band structure of periodic media and waveguides
    Physical Review B: Condensed Matter and Materials Physics, 2018
    Co-Authors: Vincent Laude, Maria Korotyaeva
    Abstract:

    We introduce a stochastic excitation method for calculating the dispersion relation for waves propagating in periodic media or along waveguides and subject to material loss or radiation damping. Instead of looking for an explicit or implicit functional relation between frequency ω and wave number k, as is usually done, we consider a mapping of the Resolvent Set in the dispersion space (ω,k). Bands appear as the trace of Lorentzian responses containing local information on propagation loss in both time and space domains. For illustration purposes, the method is applied to a lossy sonic crystal, a radiating surface phononic crystal, and a radiating optical waveguide. The Resolvent band structure can be obtained for any system described by a time-harmonic wave equation.