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.     

Exponential decay domain of energy for wave equation under feedback control

1.IntroductionLetnbeaboundedopendomaininH"(n22)withsmoothboundaryr,u~u(s)betheouternormalofrat8.GivenavectorxoEE",denotem(x)=x--xo(xER"),R=sup{im(x)IIxEfi}.SetWeconsiderthefollowingwaveequstiononnwithboundaryfeedback:whereK=(m'u)k(8),L=f(m'u),with(,n...     

具有局部非线性反馈的非均质Timoshenko梁的能量衰减估计

该文考虑具有局部非线性反馈的非均质Timoshenko梁的能量衰减估计.由非线性算子半群理论得到系统的适定性;应用乘子方法,给出了系统的能量衰减估计.     

Euler-Bernoulli梁边界反馈控制系统的Riesz基生成问题

本文用基扰动的方法,证明了由速度和角速度组成的边界反馈Ｅｕｌｅｒ－Ｂｅｒｎｏｕｌｌｉ梁振动系统的广义本征元生成状态空间Ｈ的Ｒｉｅｓｚ基,从而给出了振动系统最优指数衰减率的计算公式,     

Stabilization of transmission coupled wave and Euler–Bernoulli equations on Riemannian manifolds by nonlinear feedbacks

We consider the transmission system of coupling wave equations with Euler–Bernoulli equations on Riemannian manifolds. By introducing nonlinear boundary feedback controls, we establish the exponential and rational energy decay rate for the problem. Our proofs rely on the geometric multiplier method.     

Global solvability and decay estimates for a type III thermo-viscoelastic coupled system with infinite memory and boundary interaction feedback

A type III thermo-viscoelastic coupled system with infinite memory and distributed delay is considered. The interaction feedback between the nonlinear damping and the acoustic conditions are reacted on portion of the boundary. We obtain the well posedness and regularity of the system by using semigroup theory which is combined with Schauder's fixed point theorem. Moreover, the general decay estimates are established under a much larger class of relaxation functions. Our results are obtained without the boundedness condition of initial data assumed in many earlier papers in the literature. This work generalizes the composite stability between infinite memory and nonlinear damping.     

Energy decay of variable-coefficient wave equation with acoustic boundary conditions and delay

In this paper, we consider a variable-coefficient wave equation with memory type acoustic boundary conditions and a constant time delay in the boundary feedback. Using the Riemannian geometry method, we prove the exponential decay of the system with memory type acoustic boundary conditions and a constant time delay under some suitable assumptions.     

STABILIZATION OF VIBRATING BEAM BY VELOCITY FEEDBACK CONTROL

1IntroductionInrecentyearstherehasbeenmuchinterestintopicofcontrolandstabilizationofflexiblevibratingsystemdescribedbyaEuler-Bernoullibeamequationasfollowing(See[1]-[8]).Thequestionofstabilizationofsystem(1.O)hasbeenstudiedbymanyauthors.Forexample,seeLagnese[1],Chen.et.al[2],R.b.,b.,l3]forstabilization,Lagllese[5]forconcentratedS/A'sstabilization.Letusmentionthatthesepapersstudyasymptoticoruniformdecayforthecollsideredsystem,butnotprovetheoptimalityofthedecayrate.C..,.d[8]studytheoptimali…     

Boundary stabilization of wave equations with variable coefficients

The aim of this paper is to obtain the exponential energy decay of the solution of the wave equation with variable coefficients under suitable linear boundary feedback. Multiplier method and Riemannian geometry method are used.     

On porous-elastic systems with Fourier law

In this work, we study the asymptotic behavior of a porous elastic system coupled with the Fourier law. We show that the norm of resolvent operator is limited uniformly along the imaginary axis and we deduce that if the wave propagation speed are equal, then the system achieves exponential stability. On the other hand, if wave propagation speeds are different, then we show that the resolvent operator is not limited uniformly along the imaginary axis. This leads us to conclude that in general the model is polynomially stable.     

Global existence of solutions to a viscoelastic non-degenerate Kirchhoff equation

ABSTRACT

In this paper, a nonlinear viscoelastic kirchhoff equation in a bounded domain with a time varying delay in the weakly nonlinear internal feedback is considered, where the global existence of solutions in suitable Sobolev spaces by means of the energy method combined with Faedo–Galarkin procedure is proved with respect to the condition of the weight of the delay term in the feedback and the weight of the term without delay and the speed of delay. Furthermore, a general stability estimate using some properties of convex functions is given.     

Asymptotic Behavior for a Viscoelastic Wave Equation with a Time-varying Delay Term

The following viscoelastic wave equation with a time-varying delay term in internal feedback $|u_t|^ρu_{tt}-Δu-Δu_{tt}+∫^t_0g(t-s)Δu(s)ds+μ_1u_t(x,t)+μ_2u_t(x,t-τ(t))=0$, is considered in a bounded domain. Under appropriate conditions on μ_1, μ_2 and on the kernel g, we establish the general decay result for the energy by suitable Lyapunov functionals.     

Stabilization of elasticity–viscoporosity system by linear boundary feedback

The initial boundary value problem for linear elastodynamic system for viscoporous materials is considered. Exponential decay of solutions via the linear boundary feedback is established. Existence of solutions is obtained through the method of c₀‐semigroups. Exponential stabilization is derived via a proper collection of ideas of observability inequality, energy identity and c₀‐semigroup of contractions. Copyright © 2008 John Wiley & Sons, Ltd.     

General decay of solutions in one-dimensional porous-elastic system with memory

In this work we consider a one-dimensional porous-elastic system with memory effects. It is well-known that porous-elastic system with a single dissipation mechanism lacks exponential decay. In contrary, we prove that the unique dissipation given by the memory term is strong enough to exponentially stabilize the system, depending on the kernel of the memory term and the wave speeds of the system. In fact, we prove a general decay result, for which exponential and polynomial decay results are special cases. Our result is new and improves previous results in the literature.     

Stabilization of Euler-Bernoulli plate equation with variable coefficients by nonlinear boundary feedback

The aim of this paper is to investigate the uniform stabilization of Euler-Bernoulli plate equation with variable coefficients in the principle part subject to nonlinear boundary feedback laws. The exponential or rational energy decay rate is obtained by the multiplier method and the Riemannian geometry method.     

Well-posedness and asymptotic stability to a laminated beam in thermoelasticity of type III

This paper is concerned with the well-posedness and asymptotic behaviour of solutions to a laminated beam in thermoelasticity of type III. We first obtain the well-posedness of the system by using semigroup method. We then investigate the asymptotic behaviour of the system through the perturbed energy method. We prove that the energy of system decays exponentially in the case of equal wave speeds and decays polynomially in the case of nonequal wave speeds. Under the case of nonequal wave speeds, we also investigate the lack of exponential stability of the system.     

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.     

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.     

General energy decay for a degenerate viscoelastic Petrovsky-type plate equation with boundary feedback

In this paper, we consider a degenerate viscoelastic Petrovsky-type plate equation \[ K(\mbox{\boldmath $x$})u_{tt}+\Delta^2u-\int_0^tg(t-s)\Delta^2u(s)ds+f(u)=0 \] with boundary feedback. Under the weaker assumption on the relaxation function, the general energy decay is proved by priori estimates and analysis of Lyapunov-like functional. The exponential decay result and polynomial decay result in some literature are special cases of this paper.     

Stability to weakly dissipative Timoshenko systems

In this paper, we consider the Timoshenko systems with frictional dissipation working only on the vertical displacement. We prove that the system is exponentially stable if and only if the wave speeds are the same. On the contrary, we show that the Timoshenko systems is polynomially stable giving the optimal decay rate. Copyright © 2013 John Wiley & Sons, Ltd.