Zhuangyi 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.

A note on the equations of a thermoelastic plate

Abstract It is proved that the equations of a linear thermoelastic plate are associated with an analytic semigroup.

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.

Decay rates for a beam with pointwise force and moment feedback

Abstract. We consider the Rayleigh beam equation and the Euler–Bernoulli beam equation with pointwise feedback shear force and bending moment at the position ξ in a bounded domain (0,π) with certain boundary conditions. The energy decay rate in both cases is investigated. In the case of the Rayleigh beam, we show that the decay rate is exponential if and only if ξ/π is a rational number with coprime factorization ξ/π=p/q, where q is odd. Moreover, for any other location of the actuator we give explicit polynomial decay estimates valid for regular initial data. In the case of the Euler–Bernoulli beam, even for a nonhomogeneous material, exponential decay of the energy is proved, independently of the position of the actuator.

Exponential Stability of an Abstract Nondissipative Linear System

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

Uniform Exponential Stability and Approximation in Control of a Thermoelastic System

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

Approximations of thermoelastic and viscoelastic control systems

on a Hilbert space

Polynomial stability of a joint-leg-beam system with local damping

{\cal H}