Kangsheng Liu

Exponential Decay of Energy of the Euler--Bernoulli Beam with Locally Distributed Kelvin--Voigt Damping

In this paper, we consider the longitudinal and transversal vibrations of the Euler--Bernoulli beam with Kelvin--Voigt damping distributed locally on any subinterval of the region occupied by the beam. We prove that the semigroup associated with the equation for the transversal motion of the beam is exponentially stable, although the semigroup associated with the equation for the longitudinal motion of the beam is not exponentially stable. Due to the locally distributed and unbounded nature of the damping, we use a frequency domain method and combine a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent. We also show that the associated semigroups are not analytic.

Spectrum and stability for elastic systems with global or local Kelvin-Voigt damping

In this paper, we study the mathematical properties of a variational second order evolution equation, which includes the equations modelling vibrations of the Euler--Bernoulli and Rayleigh beams with the global or local Kelvin--Voigt (K--V) damping. In particular, our results describe the semigroup setting, the strong asymptotic stability and exponential stability of the semigroup, the analyticity of the semigroup, as well as characteristics of the spectrum of the semigroup generator under various conditions on the damping. We also give an example to show that the energy of a vibrating string does not decay exponentially when the K--V damping is distributed only on a subinterval which has one end coincident with one end of the string.

Exponential Stability of an Abstract Nondissipative Linear System

In this paper we consider an abstract linear system with perturbation of the form

On the type of C 0 -semigroup associated with the abstract linear viscoelastic system

\frac{dy}{dt}= Ay + \varepsilon By

Exponential decay of energy of vibrating strings with local viscoelasticity

on a Hilbert space

Exponential stability and analyticity of abstract linear thermoelastic systems

{\cal H}

Stability for the Timoshenko Beam System with Local Kelvin–Voigt Damping

, where A is skew-adjoint, B is bounded, and

Analyticity and Differentiability of Semigroups Associated with Elastic Systems with Damping and Gyroscopic Forces

\varepsilon