Timoshenko System Research Articles

Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term

Abstract This article concerns linear one-dimensional thermoelastic Timoshenko system with memory and distributed delay terms where the Cattaneo law governs the heat flux q ( x , t ) q\left(x,t) . We proved an exponential stability result by using the energy method combined with Lyapunov functional.

New stability result of a partially dissipative viscoelastic Timoshenko system with a wide class of relaxation function

This paper is concerned with the following partially dissipative viscoelastic Timoshenko system with damping mechanism acting only on the shear force, and with Dirichlet boundary conditions. We consider a very general relaxation function Under appropriate conditions on ξ and H, we establish a general stability result. The result is obtained under the assumption of equal speed of wave propagation. This work extends and generalizes many results in literature such as Alves et al. [On modeling and uniform stability of a partially dissipative viscoelastic Timoshenko system. SIAM J Math Anal. 2019;51(6):4520–4543].

Indirect stabilization of coupled abstract evolution equations with memory: Different propagation speeds

We consider the Cauchy problem for coupled second‐order evolution equations in a Hilbert space with indirect memory damping, which arises from the theory of viscoelasticity. Here, the coefficients of the main operators in the system are non‐equal (the case of different speeds of propagation). Under much weaker assumptions on memory kernels, we obtain a polynomial decay rate, whereas it is known that exponential stability does not hold. This work essentially generalizes and improves the related ones in the literature. As an application of our abstract results, we also establish for the Timoshenko system the so far best decay rate under such weak assumptions on the memory kernel in the case of different speeds of propagation.

Thermoelastic Timoshenko system free of second spectrum

In this work we prove the well-posedness and establish the uniform stability of a thermoelastic Timoshenko system free of second spectrum. The heat conduction is governed by classical Fourier’s law. Our stability result holds for any parameters of the system.

A transmission problem for the Timoshenko system with one local Kelvin–Voigt damping and non-smooth coefficient at the interface

In this paper, we study the indirect stability of Timoshenko system with local or global Kelvin–Voigt damping, under fully Dirichlet or mixed boundary conditions. Unlike Zhao et al. (Acta Mathematica Sinica Engl Ser 21(3):655–666, 2004), Tian and Zhang (Mathematik und Physik 68(1), 2017), and Liu and Zhang (SIAM J Control Optim 56(6):3919–3947, 2018), in this paper, we consider the Timoshenko system with only one locally or globally distributed Kelvin–Voigt damping D [see System (1.1)]. Indeed, we prove that the energy of the system decays polynomially of type \(t^{-1}\) and that this decay rate is in some sense optimal. The method is based on the frequency domain approach combining with multiplier method.

Recovering density for the Mindlin–Timoshenko system by means of a single boundary measurement

In this paper, we consider an inverse problem for the Mindlin–Timoshenko plate system, which is a strongly coupled two-dimensional system consisting of a wave equation and a system of isotropic elasticity, that arises in modeling plate vibrations especially at high frequencies and thicker plates. More precisely, we prove the global uniqueness of recovering the plate density from a single boundary measurement under appropriate geometrical assumptions. Our approach relies on diagonalizing the principal part of the system and making it a system of wave-like equations.

Dynamics of a new thermoelastic Timoshenko system with second sound

This work studies the optimal and general stability of a thermoelastic Timoshenko system with second sound. More succinctly, under some assumptions on the parameters of the system, we derive an optimal and general decay result for the solution energy of a thermoelastic Timoshenko system in which there is viscoelasticity in the transverse displacement. Examples are given to show that, polynomial and exponential decay rates are particular cases.

Uniform stability for a semilinear non-homogeneous Timoshenko system with localized nonlinear damping

Uniform stability for a semilinear non-homogeneous Timoshenko system with localized nonlinear damping

Long-time behaviour of a translational thermoelastic Timoshenko system with second sound

We consider a Timoshenko system coupled with heat equations modelled by Cattaneo's law. The coupling is through the transverse displacement. Both ends of the beam are dynamic. One end of the beam is fixed to a base in a translational motion and a tip mass is attached to the other end. We design a feedback control acting at the base. It is shown that this feedback control is a reasonable one and is capable of stabilizing the system. We prove an exponential and a polynomial stability result using the multiplier technique. To this end, we introduce new functionals to form a suitable Lyapunov functional.

A general stability result for swelling porous elastic media with nonlinear damping

We consider a swelling porous-elastic system with a single nonlinear damping in the elastic equation. Recently, Ramos et al. [Stability results for elastic porous media swelling with nonlinear damping. J Math Phys. 2020;61(10):101505.] considered the same system and established a general decay result provided that the wave speeds of the system are equal. In this paper, we obtain the general decay result without imposing a condition on the wave speeds of the system. This is a striking and unexpected result compared to Timoshenko system, porous systems, and Laminated beams system with similar damping. We also perform some numerical tests to illustrate our theoretical results.

Adaptive stabilization of a Timoshenko system by boundary feedback controls

In this work, we investigate the global existence and boundary adaptive stabilization of an undamped Timoshenko beam equation subject to two control inputs at the right end. High gain adaptive regulators are designed based on the velocity feedbacks measured at the right end. We prove an exponential stability result for the system using the multiplier method.

GLOBAL EXISTENCE OF TIMOSHENKO SYSTEM WITH RESPECT TO FRACTIONAL MEMORY OPERATOR, SPATIAL FRACTIONAL THERMAL EFFECT AND DISTRIBUTED DELAY

In this paper, the Timoshenko system with distributed delay term, fractional operator in the memory and spatial fractional thermal effect is considered, we will prove under some assumptions the global existence of a weak solution. Furthermore, we show some results about the stability of system by the semigroup method.

General decay for a viscoelastic-type Timoshenko system with thermoelasticity of type III

We consider a one-dimensional thermoelastic Timoshenko system, where the heat conduction is given by Green and Naghdi theories and acting on shear force. We prove that the unique damping given by the memory is sufficiently strong to stabilize the system exponentially, depending on a new relationship between the coefficients of the system and under some assumptions on the kernel of the memory term. In fact, we establish a general decay result, of which exponential and polynomial decay results are special cases.

Stability conditions for thermodiffusion Timoshenko system with second sound

Stability conditions for thermodiffusion Timoshenko system with second sound

Exponential stabilization of a Timoshenko system with thermodiffusion effects

Exponential stabilization of a Timoshenko system with thermodiffusion effects

On the control of viscoelastic damped swelling porous elastic soils with internal delay feedbacks

We consider a swelling porous-elastic system with viscoelastic damping and delay feedbacks acting on the fluid equation. Using the multiplier method and under the well-known assumption on the weight of delay term, we unexpectedly establish a general decay result without imposing the usual condition of equal wave speeds of the system, unlike the case of Timoshenko and porous systems where damping on only one of the equations requires equal wave speeds propagation. Our coupling and the result give new contributions to the theory associated with asymptotic behaviors of swelling porous elastic soils.

Indirect stabilization of Timoshenko systems with one local memory damping

We investigate the indirect stabilization of Timoshenko systems with only one local memory effect. By the multiplier method and some specially designed analysis approaches for the problem, we obtain the exponential decay rates for the case in which the wave speeds are the same. Of particular notes are: This is a local memory damping problem for coupled systems without complementary frictional damping (or any other type of damping); It is also an indirect damping problem, which has memory damping only in the second equation and no damping in the first equation. The decay results here are established without the usual nonnegative/decreasing assumptions for the kernel, that is, the kernel may be oscillating or sign-varying. As can be seen, we weaken the hypotheses of the kernel and the memory-effect area, but we still establish the exponential decay rates of the energy. Therefore, we have improved and extended the related results to a considerable extent.

Exponential stability in a Timoshenko system of type III

In this paper, a Timoshenko system with a delay term and a thermo-viscoelastic damping of type III subjected to Dirichlet–Dirichlet–Neumann boundary conditions will be considered. Exploiting energy method to produce a suitable Lyapunov functional, we establish the global existence and exponential decay of energy for Timoshenko system of type III.

Energy decay for damped Shear beam model and new facts related to the classical Timoshenko system

In this paper, we consider a beam model known as Shear beam model (no rotary inertia). We assure, based on behavior of the wave speeds, that the Shear beam model corresponds to a part of the classical Timoshenko beam model which is governed only by one wave speed. Unlike the classical dissipative Timoshenko type system with viscous damping acting on transverse displacement, we prove that the corresponding dissipative Shear beam model has an energy exponential decay regardless any relationship between coefficients of the system. This happens because such model has only one finite wave speed for all wave numbers.

Exponential stability of thermoelastic Timoshenko system with Cattaneo’s law

In this paper, we consider a one dimensional thermoelastic Timoshenko system where the thermal coupling is acting on both the shear force and the bending moment, and the heat flux is given by Cattaneo’s law. We establish an exponential stability result irrespective of the values of the coefficients of the system.