A stability and numerical study of the solutions of a Timoshenko system with distributed delay

In this work, we analyze the stability of the semigroup associated with a Timoshenko beam model with distributed delay in the rotation angle equation. We show that the type of stability resulting from the semigroup is directly related to some model coefficients, which constitute the velocities of the system's component equations. In the case of stability of the polynomial type, we prove that rate obtained is optimal. We conclude the work performing a numerical study of the solutions and their energies, associated to discrete system.     

Stability of a thermoelastic laminated system subject to a neutral delay

In this work, we consider a system of two identical beams of uniform thickness modeled as a Timoshenko system. The slip between the beams is taken into account, and the system is coupled with a heat equation. Moreover, the slip equation is subject to a distributed delay of neutral type. Delays are known to be of a destructive nature in general. Therefore, here, the delay will compete with a frictional damping and the dissipation produced by the heat equation. We provide sufficient conditions ensuring exponential and polynomial stability of the structure.     

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 asymptotic behavior of a wave equation with distributed time‐delay and Neumann boundary conditions

This paper is concerned with the asymptotic behavior analysis of solutions to a multidimensional wave equation. Assuming that there is no displacement term in the system and taking into consideration the presence of distributed or discrete time delay, we show that the solutions exponentially converge to their stationary state. The proof mainly consists in utilizing the resolvent method. The approach adopted in this work is also used to other physical systems.     

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.     

On a size-structured population model with infinite states-at-birth and distributed delay in birth process

In this article, we study a size-structured population model with infinite states-at-birth and distributed delay in birth process. We establish the well-posedness for this model and show that the solution of this model exhibits an asynchronous exponential growth by means of semigroups.     

Polynomial and exponential stability for one-dimensional problem in thermoelastic diffusion theory

We consider one-dimensional problem for the thermoelastic diffusion theory and we obtain polynomial decay estimates. Then we show that the solution decays exponentially to zero as time goes to infinity; that is, denoting by E(t) the first-order energy of the system, we show that positive constants C ₀ and c ₀ exist which satisfy E(t) ≤ C ₀ E(0)e ^{?c ₀ t} .     

Asymptotical stability for memory‐type porous thermoelastic system of type III with constant time delay

In this paper, we consider a one‐dimensional porous thermoelastic system of type III with viscoelastic damping and constant time delay on boundary. Using the energy method, we prove the general stability of the system under suitable assumptions on the relaxation function and the time delay. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Uniform stability for thermoelastic systems with boundary time-varying delay

In this paper we consider a thermoelastic system with boundary time-varying delay. Using the energy method, we show, under suitable assumptions, that the damping effect through heat conduction is still strong enough to uniformly stabilize the system even in the presence of boundary time-varying delay. Our result improves earlier results existing in the literature.     

A uniform stability result for thermoelasticity of type III with boundary distributed delay

In this paper we consider a thermoelastic system of type III with boundary distributed delay. Under suitable assumption on the weight of the delay, we prove, using the energy method, that the damping effect through heat conduction given by Green and Naghdi's theory is still strong enough to uniformly stabilize the system even in the presence of time delay.     

Stabilization of a viscoelastic Timoshenko beam

A viscoelastic Timoshenko beam is investigated. We prove an exponential decay of solutions for a large class of kernels with weaker conditions than the existing ones in the literature. This will allow the use of other types of viscoelastic material for Timoshenko type beams than the usually used ones.     

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.     

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.     

New general and explicit stability result for a thermoelastic Timoshenko system

Ricerche di Matematica - The goal of this article is to study a thermoelastic Timoshenko system with viscoelastic law acting on the shear force and thermoelastic dissipation acting on the bending...     

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．     

Decay rates for memory‐type plate system with delay and source term

In this paper, we consider a plate equation with infinite memory in the presence of delay and source term. Under suitable conditions on the delay and source term, we establish an explicit and general decay rate result without imposing restrictive assumptions on the behavior of the relaxation function at infinity. Our result allows a wider class of relaxation functions and improves earlier results in the literature. Copyright © 2016 John Wiley & Sons, Ltd.     

The exact boundaries of the stability domains of linear differential equations with distributed delay

For a differential equation with a distributed varying delay, sufficient criteria for the asymptotic and uniform stability of solutions are obtained. The constructed examples demonstrate exactness of the boundary of the obtained stability domain.