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On Spectrum and Riesz basis property for one-dimensional wave equation with Boltzmann damping∗

Bao-Zhu Guo, Guo-Dong Zhang (2012)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we study the one-dimensional wave equation with Boltzmann damping. Two different Boltzmann integrals that represent the memory of materials are considered. The spectral properties for both cases are thoroughly analyzed. It is found that when the memory of system is counted from the infinity, the spectrum of system contains a left half complex plane, which is sharp contrast to the most results in elastic vibration systems that the vibrating dynamics can be considered from the vibration...

On the basis property of the root functions of differential operators with matrix coefficients

Oktay Veliev (2011)

Open Mathematics

We obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and periodic or antiperiodic boundary conditions. Then using these asymptotic formulas, we find necessary and sufficient conditions on the coefficients for which the system of eigenfunctions and associated functions of the operator under consideration forms a Riesz basis.

On the canonical development of Parseval formulas for singular differential operators

Robert W. Carroll (1982)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Per funzioni opportune f , g si ottiene una formula di Parseval ⟨ 𝐑 Q , 𝒬 ⁢ f ⁢ 𝒬 ⁢ g ⟩ λ = ⟨ Δ Q - 1 / 2 ⁢ f , Δ Q - 1 / 2 ⁢ g ⟩ per operatori differenziali singolari di tipo dell'operatore radiale di Laplace-Beltrami. 𝐑 Q è una funzione spettrale generalizzata di tipo Marčenko e può essere rappresentata per mezzo di un certo nucleo della trasmutazione.

On the dynamic behavior and stability of controlled connected Rayleigh beams under pointwise output feedback

Bao-Zhu Guo, Jun-Min Wang, Cui-Lian Zhou (2008)

ESAIM: Control, Optimisation and Calculus of Variations

We study the dynamic behavior and stability of two connected Rayleigh beams that are subject to, in addition to two sensors and two actuators applied at the joint point, one of the actuators also specially distributed along the beams. We show that with the distributed control employed, there is a set of generalized eigenfunctions of the closed-loop system, which forms a Riesz basis with parenthesis for the state space. Then both the spectrum-determined growth condition and exponential stability...

On the solvability of the Lyapunov equation for nonselfadjoint differential operators of order 2m with nonlocal boundary conditions

Aris Tersenov (2001)

Annales Polonici Mathematici

This paper is devoted to the solvability of the Lyapunov equation A*U + UA = I, where A is a given nonselfadjoint differential operator of order 2m with nonlocal boundary conditions, A* is its adjoint, I is the identity operator and U is the selfadjoint operator to be found. We assume that the spectra of A* and -A are disjoint. Under this restriction we prove the existence and uniqueness of the solution of the Lyapunov equation in the class of bounded operators.

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