Well posedness and stability result for a thermoelastic laminated Timoshenko beam with distributed delay term

In this work, we consider a linear thermoelastic laminated timoshenko beam with distributed delay, where the heat conduction is given by cattaneoâs law. we establish the well posedness of the system. For stability results, we prove exponential and polynomial stabilities of the system for the cases of equal and nonequal speeds of wave propagation.     

Stabilization of a type III thermoelastic Timoshenko system in the presence of a time‐distributed delay

In this paper, we investigate a Timoshenko‐type system of thermoelasticity of type III in the presence of a distributed delay. We prove the well‐posedness and two exponential stability results in the presence as well as in the absence of an extra frictional damping. In case of absence of the frictional damping, our stability result is obtained under the equal‐speed of propagation and a smallness condition on the weight of delay.     

Long-time dynamics for a nonlinear Timoshenko system with delay

This paper is concerned with a nonlinear Timoshenko system with a time delay term in the internal feedback together with initial data and Dirichet boundary conditions. Under some suitable assumptions on the weights of feedback, we obtain the existence of a global attractor with finite fractal dimension for the case of equal speed wave propagation. Furthermore, the existence of exponential attractors is also derived.     

Stabilization of the uniform Timoshenko beam by one locally distributed feedback

In this article, we study the energy decay rate for an elastic Timoshenko system. This system consists of two coupled wave equations. Only the equation about the rotation angle is damped by one locally distributed feedback at the neighbourhood of the boundary. The equation for the transverse displacement of the beam is only indirectly damped through the coupling. First, we establish an exponential energy decay rate in the case of the same speed of propagation. Next, when the wave speeds are different, a polynomial-type decay rate is obtained. These results are proved by verifying the frequency domain conditions.     

A stability result of a Timoshenko system with a delay term in the internal feedback

In this paper, we consider a Timoshenko system with a delay term in the feedback and prove a stability result. The beam is clamped at the endpoints and has, in addition to an internal damping, a feedback with a delay.Under an appropriate assumption on the weights of the two feedbacks, we prove the well-posedness of the system and establish an exponential decay result for the case of equal-speed wave propagation.     

About partial boundary dissipation to Timoshenko system with delay

We consider the Timoshenko model with partial dissipative boundary condition with delay, and we prove that the solution decays exponentially to zero, provided the wave speed are equal; this improve earlier result due to Bassam et al and Muñoz Rivera and Naso. Moreover, consider the exponential stability to the corresponding semilinear problems.     

具有内部点耗散的Timoshenko梁的能量衰减估计

研究具有反馈控制力的Timoshenko梁的能量衰减．证明了梁的能量不是一致衰减的．当梁的能量不是一致衰减时,利用初始值的正则性和无阻尼问题的最佳正则性结果,给出了多项式衰减估计．     

Uniform decay of energy for a porous thermoelasticity system with past history

In this paper, we consider a one-dimensional porous thermoelasticity system with past history, which contains a porous elasticity in the presence of a visco-porous dissipation, a macrotemperature effect and temperature difference. We establish the exponential stability of the system if and only if the equations have the same wave speeds, and obtain the energy decays polynomially to zero in the case that the wave speeds of the equations are different.     

Exponential Decay Rate for a Timoshenko Beam with Boundary Damping

The exponential decay rate of a Timoshenko beam system with boundary damping is studied. By asymptotically analyzing the characteristic determinant of the system, we prove that the Timoshenko beam system is a Riesz system; hence, its decay rate is determined via its spectrum. As a consequence, by showing that the imaginary axis neither has an eigenvalue on it nor is an asymptote of the spectrum, we conclude that the system is exponentially stable.     

A general stability result in a Timoshenko system with infinite memory: A new approach

In this paper, we consider a Timoshenko system in the presence of an infinite memory, where the relaxation function satisfies a relation of the form Under the same hypothesis on g and ξ imposed for the finite memory case, we establish some general decay results for the equal and nonequal speed propagation cases. Our results improve in some situations some known decay rates. Also, some applications to other problems are discussed. Copyright © 2013 John Wiley & Sons, Ltd.     

Energy decay to Timoshenko system with indefinite damping

We consider the classical Timoshenko system for vibrations of thin rods. The system has an indefinite damping mechanism, ie, it has a damping function a=a(x) possibly changing sign, present only in the equation for the vertical displacement. We shall prove that exponential stability depends on conditions regarding of the indefinite damping function a and a nice relationship between the coefficient of the system. Finally, we give some numerical result to verify our analytical results.     

Exponential decay of Timoshenko beam with locally distributed feedback

The problem of exponential stabilization of a nonuniform Timoshenkobeam with locally distributed controls is investigated. Withoutthe assumption of different wave speeds, it is shown that, undersome locally distributed controls, the vibration of the beamdecays exponentially. The proof is obtained by using a frequencymultiplier method.     

Stability of Timoshenko systems with past history

We consider vibrating systems of Timoshenko type with past history acting only in one equation. We show that the dissipation given by the history term is strong enough to produce exponential stability if and only if the equations have the same wave speeds. Otherwise the corresponding system does not decay exponentially as time goes to infinity. In the case that the wave speeds of the equations are different, which is more realistic from the physical point of view, we show that the solution decays polynomially to zero, with rates that can be improved depending on the regularity of the initial data.     

A numerical algorithm for the nonlinear Timoshenko beam system

An initial boundary value problem is considered for the dynamic beam system Its solution is found by means of an algorithm, the constituent parts of which are the finite element method, the implicit symmetric difference scheme used to approximate the solution with respect to the spatial and time variables, and also a Picard type iteration process for solving the system of nonlinear equations obtained by discretization. Errors of three parts of the algorithm are estimated and, as a result, its total error estimate is obtained. A numerical example is solved.     

A boundary obstacle problem for the Mindlin–Timoshenko system

We consider the dynamical one‐dimensional Mindlin–Timoshenko model for beams. We study the existence of solutions for a contact problem associated with the Mindlin–Timoshenko system. We also analyze how its energy decays exponentially to zero as time goes to infinity. Copyright © 2008 John Wiley & Sons, Ltd.     

Nonlinear damped Timoshenko systems with second sound—Global existence and exponential stability

In this paper, we consider nonlinear thermoelastic systems of Timoshenko type in a one‐dimensional bounded domain. The system has two dissipative mechanisms being present in the equation for transverse displacement and rotation angle—a frictional damping and a dissipation through hyperbolic heat conduction modelled by Cattaneo's law, respectively. The global existence of small, smooth solutions and the exponential stability in linear and nonlinear cases are established. Copyright © 2008 John Wiley & Sons, Ltd.     

The fixed sign property of solutions and stability of linear differential equations with varying distributed delay

We study one class of linear differential equations with varying distributed delay. We obtain an effective (in terms of parameters of the initial problem) criterion for the positivity of the Cauchy function of this class of equations. Based on this result, we establish effective criteria for the exponential stability of equations under consideration.     

Energy decay rates for a Timoshenko-type system of thermoelasticity of type III with constant delay

In this paper, we consider a one-dimensional linear thermoelastic system of Timoshenko type with delay, where the heat conduction is given by Green and Naghdi theory. We establish the stability of the system for the case of equal and nonequal speeds of wave propagation.     

On the stabilization of the Timoshenko system by a weak nonlinear dissipation

In this paper we consider the following Timoshenko‐type system: Without imposing any restrictive growth assumption on g at the origin, we establish a general decay result depending on g and α. Copyright © 2008 John Wiley & Sons, Ltd.