scholarly journals On the stability of a laminated beam with structural damping and Gurt–Pipkin thermal law

2021 ◽  
Vol 26 (3) ◽  
pp. 396-418
Author(s):  
Wenjun Liu ◽  
Weifan Zhao
Keyword(s):  
Exponential Stability ◽  
Relaxation Function ◽  
Equation Modeling ◽  
Well Posedness ◽  
Structural Damping ◽  
Stability Number ◽  
Laminated Beam ◽  
One Dimensional ◽  
Lack Of Exponential Stability ◽  
The Stability

In this paper, we investigate the stabilization of a one-dimensional thermoelastic laminated beam with structural damping coupled with a heat equation modeling an expectedly dissipative effect through heat conduction governed by Gurtin–Pipkin thermal law. Under some assumptions on the relaxation function g, we establish the well-posedness of the problem by using Lumer–Phillips theorem. Furthermore, we prove the exponential stability and lack of exponential stability depending on a stability number by using the perturbed energy method and Gearhart–Herbst–Prüss–Huang theorem, respectively.

scholarly journals On the Stability of a Laminated Beam With Structural Damping and Gurtin-Pipkin Thermal Law

2018 ◽  
Author(s):  
Wenjun Liu ◽  
Weifan Zhao
Keyword(s):  
Exponential Stability ◽  
Relaxation Function ◽  
Equation Modeling ◽  
Well Posedness ◽  
Structural Damping ◽  
Laminated Beam ◽  
One Dimensional ◽  
Semigroup Method ◽  
Lack Of Exponential Stability ◽  
The Stability

In this paper, we investigate the stabilization of a one-dimensional thermoelastic laminated beam with structural damping, coupled to a heat equation modeling an expectedly dissipative effect through heat conduction governed by Gurtin-Pipkin thermal law. Under some assumptions on the relaxation function g, we establish the well-posedness for the problem. Furthermore, we prove the exponential stability and lack of exponential stability for the problem. To achieve our goals, we make use of the semigroup method, the perturbed energy method and Gearhart-Herbst-Prüss-Huang theorem.

scholarly journals Uniform stability for a type III thermoelastic laminated beam with structural damping

2022 ◽  
Author(s):  
Fayssal Djellali
Keyword(s):  
Heat Flux ◽  
Exponential Stability ◽  
Semigroup Theory ◽  
Stability Result ◽  
Uniform Stability ◽  
Well Posedness ◽  
Structural Damping ◽  
Laminated Beam ◽  
Wave Speeds ◽  
Lack Of Exponential Stability

In this work, we consider a thermoelastic laminated beam with structural damping, where the heat flux is given by Green and Naghdi theories. We establish the well-posedness of the system using semigroup theory. Moreover, under the condition of equal wave speeds, we prove an exponential stability result for the considered system. In the case of lack of exponential stability we show that the solution decays polynomially.

Asymptotic stability for a laminated beam with structural damping and infinite memory

2020 ◽  
Vol 25 (10) ◽  
pp. 1979-2004 ◽  
Author(s):  
Wenjun Liu ◽  
Xiangyu Kong ◽  
Gang Li
Keyword(s):  
Wave Propagation ◽  
Relaxation Function ◽  
Rotation Angle ◽  
Structural Damping ◽  
Laminated Beam ◽  
Infinite Memory ◽  
One Dimensional ◽  
Special Cases ◽  
Lack Of Exponential Stability ◽  
Well Posed

In this paper, we consider a one-dimensional laminated beam with structural damping and an infinite memory acting on the effective rotation angle. Under appropriate assumptions imposed on the relaxation function, we show that the system is well-posed by using the Hille–Yosida theorem, and then we establish general decay results, from which exponential and polynomial decays are only special cases, in the case of equal speeds of wave propagation as well as that of nonequal speeds. In the particular case when the wave propagation speeds are different and the relaxation function decays exponentially, we show the lack of exponential stability.

scholarly journals Lack of Exponential Decay for a Laminated Beam with Structural Damping and Second Sound

2017 ◽  
Author(s):  
Wenjun Liu ◽  
Xiangyu Kong ◽  
Gang Li
Keyword(s):  
Heat Conduction ◽  
Exponential Decay ◽  
Polynomial Decay ◽  
Second Sound ◽  
Structural Damping ◽  
Stability Number ◽  
Laminated Beam ◽  
One Dimensional ◽  
The Stability ◽  
Math Phys

In previous work (Z. Angew. Math. Phys. 68(2), 2017), Apalara considered a one dimensional thermoelastic laminated beam under Cattaneo’s law of heat conduction and proved the exponential and polynomial decay results depend on the stability number χT . In this paper, we continue to study the same system and show that the solution of the concerned system lacks of exponential decay result in the case χT ≠ 0 which solves the open problem proposed by Apalara (Z. Angew. Math. Phys. 68(2), 2017).

Exponential stability for a thermoelastic laminated beam with nonlinear weights and time-varying delay

2021 ◽  
pp. 1-29
Author(s):  
Carlos Nonato ◽  
Carlos Raposo ◽  
Baowei Feng
Keyword(s):  
Exponential Stability ◽  
Existence And Uniqueness ◽  
Semigroup Theory ◽  
Well Posedness ◽  
Time Varying ◽  
Structural Damping ◽  
Laminated Beam ◽  
Time Varying Delay ◽  
Timoshenko Systems ◽  
Varying Delay

In this paper, we study the well-posedness and asymptotic stability to a thermoelastic laminated beam with nonlinear weights and time-varying delay. To the best of our knowledge, there are no results on the system and related Timoshenko systems with nonlinear weights. On suitable premises about the time delay and the hypothesis of equal-speed wave propagation, existence and uniqueness of solution is obtained by combining semigroup theory with Kato variable norm technique. The exponential stability is proved by energy method in two cases, with and without the structural damping, by using suitably sophisticated estimates for multipliers to construct an appropriated Lyapunov functional.

scholarly journals EXPONENTIAL STABILITY OF LAMINATED BEAM WITH CONSTANT DELAY FEEDBACK

2021 ◽  
Vol 26 (4) ◽  
pp. 566-581
Author(s):  
Kassimu Mpungu ◽  
Tijani A. Apalara
Keyword(s):  
Wave Propagation ◽  
Exponential Stability ◽  
Propagation Speed ◽  
Transverse Displacement ◽  
Structural Damping ◽  
Constant Delay ◽  
Laminated Beams ◽  
Laminated Beam ◽  
Delay Term ◽  
Wave Propagation Speed

In this article, we consider a system of laminated beams with an internal constant delay term in the transverse displacement. We prove that the dissipation through structural damping at the interface is strong enough to exponentially stabilize the system under suitable assumptions on delay feedback and coefficients of wave propagation speed.

scholarly journals On the Porous-Elastic System with Thermoelasticity of Type III and Distributed Delay: Well-Posedness and Stability

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Djamel Ouchenane ◽  
Abdelbaki Choucha ◽  
Mohamed Abdalla ◽  
Salah Mahmoud Boulaaras ◽  
Bahri Belkacem Cherif
Keyword(s):  
Lyapunov Functions ◽  
Elastic System ◽  
Distributed Delay ◽  
Mathematical Physics ◽  
Well Posedness ◽  
Engineering Structures ◽  
One Dimensional ◽  
Delay Term ◽  
Type Iii ◽  
The Stability

The paper deals with a one-dimensional porous-elastic system with thermoelasticity of type III and distributed delay term. This model is dealing with dynamics of engineering structures and nonclassical problems of mathematical physics. We establish the well posedness of the system, and by the energy method combined with Lyapunov functions, we discuss the stability of system for both cases of equal and nonequal speeds of wave propagation.

Regularity and stability of coupled plate equations with indirect structural or Kelvin-Voigt damping

2019 ◽  
Vol 25 ◽  
pp. 51 ◽  
Author(s):  
Zhong-Jie Han ◽  
Zhuangyi Liu
Keyword(s):  
Spectral Analysis ◽  
Exponential Stability ◽  
Numerical Simulations ◽  
Frequency Domain ◽  
Frequency Domain Method ◽  
Structural Damping ◽  
One Dimensional ◽  
Domain Method ◽  
Plate Equations ◽  
Exponentially Stable

In this paper, the regularity and stability of the semigroup associated with a system of coupled plate equations is considered. Indirect structural or Kelvin-Voigt damping is imposed, i.e., only one equation is directly damped by one of these two damping. By the frequency domain method, we show that the associated semigroup of the system with indirect structural damping is analytic and exponentially stable. However, with the much stronger indirect Kelvin-Voigt damping, we prove that, by the asymptotic spectral analysis, the semigroup is even not differentiable. The exponential stability is still maintained. Finally, some numerical simulations of eigenvalues of the corresponding one-dimensional systems are also given.

scholarly journals Well-Posedness and Asymptotic Stability to a Laminated Beam in Thermoelasticity of Type III

2018 ◽  
Author(s):  
Yue Luan ◽  
Wenjun Liu ◽  
Gang Li
Keyword(s):  
Asymptotic Stability ◽  
Stability Result ◽  
Well Posedness ◽  
Polynomial Stability ◽  
Laminated Beam ◽  
Wave Speeds ◽  
Type Iii ◽  
Semigroup Method ◽  
Exponentially Stable ◽  
Lack Of Exponential Stability

In this paper, we study the well-posedness and asymptotic behaviour of solutions to a laminated beam in thermoelasticity of type III. We first give the well-posedness of the system by using the semigroup method. Then, we show that the system is exponentially stable under the assumption of equal wave speeds. Furthermore, it is proved that the system is lack of exponential stability for case of nonequal wave speeds. In this regard, a polynomial stability result is proved.

scholarly journals Exponential and Polynomial Decay for a Laminated Beam with Fourier's Type Heat Conduction

2019 ◽  
Author(s):  
Wenjun Liu ◽  
Weifan Zhao
Keyword(s):  
Heat Conduction ◽  
Energy Method ◽  
Rotation Angle ◽  
Polynomial Decay ◽  
Structural Damping ◽  
Laminated Beam ◽  
Wave Speeds ◽  
One Dimensional ◽  
Exponentially Stable ◽  
Well Posed

In this paper, we study the well-posedness and asymptotics of a one-dimensional thermoelastic laminated beam system either with or without structural damping, where the heat conduction is given by Fourier's law effective in the rotation angle displacements. We show that the system is well-posed by using Lumer-Philips theorem, and prove that the system is exponentially stable if and only if the wave speeds are equal, by using the perturbed energy method and Gearhart-Herbst-Prüss-Huang theorem. Furthermore, we show that the system with structural damping is polynomially stable provided that the wave speeds are not equal, by using the second-order energy method.

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