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.     

Stability results for laminated beam with thermo-visco-elastic effects and localized nonlinear damping

In this article, we investigate a one-dimensional thermoelastic laminated beam system with nonlinear damping and viscoelastic dissipation on the effective rotation angle and through heat conduction in the interfacial slip equations. Under minimal conditions on the relaxation function and the relationship between the coefficients of the wave propagation speed of the first two equations, we show that the solution energy has an explicit and optimal decay rate from which the exponential and polynomial stability are just particular cases. Moreover, we establish a weaker decay result in the case of non-equal wave of speed propagation and give some examples illustrate our results. This work extends and improves the earlier results in the literature, particularly the result of Mukiawa et al. (2021).     

Stabilisation faible interne locale de système élastique de Bresse

In [1], Alabau-Boussouira et al. (2011) studied the exponential and polynomial stability of the Bresse system with one globally distributed dissipation law. In this Note, our goal is to extend the results from Alabau-Boussouira et al. (2011) [1], by taking into consideration the important case when the dissipation law is locally distributed and to improve the polynomial energy decay rate. We then study the energy decay rate of the Bresse system with one locally internal distributed dissipation law acting on the equation about the shear angle displacement. Under the equal speed wave propagation condition, we show that the system is exponentially stable. On the contrary, we establish a new polynomial energy decay rate.     

Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control

We consider systems of Timoshenko type in a one-dimensional bounded domain. The physical system is damped by a single feedback force, only in the equation for the rotation angle, no direct damping is applied on the equation for the transverse displacement of the beam. Moreover the damping is assumed to be nonlinear with no growth assumption at the origin, which allows very weak damping. We establish a general semi-explicit formula for the decay rate of the energy at infinity in the case of the same speed of propagation in the two equations of the system. We prove polynomial decay in the case of different speed of propagation for both linear and nonlinear globally Lipschitz feedbacks.      

Exponential and polynomial decay for a laminated beam with Fourier's law of heat conduction and possible absence of structural damping

We study the well-posedness and decay properties of a onedimensional thermoelastic laminated beam system either with or without structural damping, of which the heat conduction is given by Fourier's law effective in the rotation angle displacements. We show that the system is wellposed by using the 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. When the speeds are not equal, whether the system without structural damping may has polynomial stability is left as an open problem.     

Asymptotic behavior of the porous elastic system with dual phase lag model: Classical versus second spectrum perspectives

This paper aims to analyze the energy decay of the thermoelastic porous system. The dual-phase lag theory is used to model heat transfer. We consider two perspectives: the classical approach and the second spectrum approach. For the classical approach, the well-posedness is obtained via the semigroup theory and the system is exponentially stable under equal wave speed conditions. On the opposite, we show a polynomial decay. On the other hand, the well-posedness of the truncated system is obtained via the Faedo Galerkin method, and the system is exponentially stable without any assumptions on the physical parameters.     

On the exponential and polynomial stability for a linear Bresse system

In this paper, we consider a linear one-dimensional Bresse system consisting of three hyperbolic equations coupled in a certain manner under mixed homogeneous Dirichlet-Neumann boundary conditions. Here, we consider that only the longitudinal displacement is damped, and the vertical displacement and shear angle displacement are free. We prove the well-posedness of the system and some exponential, lack of exponential and polynomial stability results depending on the coefficients of the equations and the smoothness of initial data. At the end, we use some numerical approximations based on finite difference techniques to validate the theoretical results. The proof is based on the semigroup theory and a combination of the energy method and the frequency domain approach.     

Stabilization of the generalized Rao-Nakra beam by partial viscous damping

In this paper, we consider the stabilization of the generalized Rao-Nakra beam equation, which consists of four wave equations for the longitudinal displacements and the shear angle of the top and bottom layers and one Euler–Bernoulli beam equation for the transversal displacement. Dissipative mechanism are provided through viscous damping for two displacements. The location of the viscous damping are divided into two groups, characterized by whether both of the top and bottom layers are directly damped or otherwise. Each group consists of three cases. We obtain the necessary and sufficient conditions for the cases in group 2 to be strongly stable. Furthermore, polynomial stability of certain orders are proved. The cases in group 1 are left for future study.     

The longtime behavior of a nonlinear Reissner-Mindlin plate with exponentially decreasing memory kernels

In this paper we investigate the longtime behavior of the mathematical model of a homogeneous viscoelastic plate based on Reissner-Mindlin deformation shear assumptions. According to the approximation procedure due to Lagnese for the Kirchhoff viscoelastic plate, the resulting motion equations for the vertical displacement and the angle deflection of vertical fibers are derived in the framework of the theory of linear viscoelasticity. Assuming that in general both Lame's functions, λ and μ, depend on time, the coupling terms between the equations of displacement and deflection depend on hereditary contributions. We associate to the model a nonlinear semigroup and show the behavior of the energy when time goes on. In particular, assuming that the kernels λ and μ decay exponentially, and not too weakly with respect to the physical properties considered in the model, then the energy decays uniformly with respect to the initial conditions; i.e., we prove the existence of an absorbing set for the semigroup associated to the model.     

Stability results of a distributed problem involving Bresse system with history and/or Cattaneo law under fully Dirichlet or mixed boundary conditions

In this paper, we study the stability of a 1‐dimensional Bresse system with infinite memory‐type control and/or with heat conduction given by Cattaneo's law acting in the shear angle displacement. When the thermal effect vanishes, the system becomes elastic with memory term acting on one equation. We consider the interesting case of fully Dirichlet boundary conditions. Indeed, under equal speed of propagation condition, we establish the exponential stability of the system. However, in the natural physical case when the speeds of propagation are different, using a spectrum method, we show that the Bresse system is not uniformly exponentially stable. In this case, we establish a polynomial energy decay rate. Our study is valid for all other mixed boundary conditions.     

Stabilization of a thermoelastic Mindlin–Timoshenko plate model revisited

This paper is a continuation of our work in Grobbelaar-Van Dalsen (Appl Anal 90:1419–1449, 2011) where we showed the strong stability of models involving the thermoelastic Mindlin–Timoshenko plate equations with second sound. For the case of a plate configuration consisting of a single plate, this was accomplished in radially symmetric domains without applying any mechanical damping mechanism. Further to this result, we establish in this paper the non-exponential stability of the model for a particular configuration under mixed boundary conditions on the shear angle variables and Dirichlet boundary conditions on the displacement and thermal variables when the heat flux is described by Fourier’s law of heat conduction. We also determine the rate of polynomial decay of weak solutions of the model in a radially symmetric region under Dirichlet boundary conditions on the displacement and thermal variables and free boundary conditions on the shear angle variables.     

Generalized thermoelasticity of beams under partial thermal shock

In this paper, the generalized thermoelastic response of a beam subjected to a partial lateral thermal shock is analysed. The beam is made of homogeneous and isotropic material and is assumed to follow the Hooke law for its constitutive material. The displacement gradient is small and the linear form of strain-displacement relations is used for the beam. The equations of motion and the boundary conditions of the beam are derived based on Hamilton’s principle. According to the first and second laws of thermodynamics, a non-Fourier constitutive equation is employed to derive the energy equation of the beam. The non-Fourier effects lead to the constitutive equation of the hyperbolic type and thus the thermal and mechanical waves can be observed. The propagation of waves in the beam are simulated by finite element model and the wave reflections for different types of boundary conditions are studied. The relaxation time is considered as a significant parameter and results show that energy absorption of the structure and the wave propagation speed depend upon this parameter.     

Uniform Decay properties of a model in structural acoustics

We investigate decay properties for a system of coupled partial differential equations which model the interaction between acoustic waves in a cavity and the walls of the cavity. In this system a wave equation is coupled to a structurally damped plate or beam equation. The underlying semigroup for this system is not uniformly stable, but when the system is appropriately restricted we obtain some uniform stability. We present two results of this type. For the first result, we assume that the initial wave data is zero, and the initial plate or beam data is in the natural energy space; then the corresponding solution to system decays uniformly to zero. For the second result, we assume that the initial condition is in the natural energy space and the control function is L²(0,∞) (in time) into the control space; then the beam displacement and velocity are both L²(0,∞) into a space with two spatial derivatives.     

Global Solution and Exponential Stability for a Laminated Beam with Fourier Thermal Law

This paper focuses on the long-time dynamics of a thermoelastic laminated beam modeled from the well-established Timoshenko theory. From mathematical point of view, the study system consists of three hyperbolic motion equations coupled with the parabolic equation governed by Fouriers law of heat conduction and, in consequence, does not belong to one of the classical categories of PDE. We have proved the well-posedness and exponential stability of the system. The well-posedness is given by Hille-Yosida theorem. For the exponential decay we applied the energy method by introducing a Lyapunov functional.     

Stability analysis of coupled structural acoustics PDE models under thermal effects and with no additional dissipation

In this study we consider a coupled system of partial differential equations (PDE's) which describes a certain structural acoustics interaction. One component of this PDE system is a wave equation, which serves to model the interior acoustic wave medium within a given three dimensional chamber Ω. This acoustic wave equation is coupled on a boundary interface Γ₀ to a two dimensional system of thermoelasticity: this thermoelastic PDE is composed in part of a structural beam or plate equation, which governs the vibrations of flexible wall portion Γ₀ of the chamber Ω. Moreover, this elastic dynamics is coupled to a heat equation which also evolves on Γ₀, and which imparts a thermal damping onto the entire structural acoustic system. As we said, the interaction between the wave and thermoelastic PDE components takes place on the boundary interface Γ₀, and involves coupling boundary terms which are above the level of finite energy. We analyze the stability properties of this coupled structural acoustics PDE model, in the absence of any additive feedback dissipation on the hard walls Γ₁ of the boundary . Under a certain geometric assumption on Γ₁, an assumption which has appeared in the literature in connection with structural acoustic flow, and which allows for the invocation of a recently derived microlocal boundary trace estimate, we show that classical solutions of this thermally damped structural acoustics PDE decay uniformly to zero, with a rational rate of decay.     

Frequency domain approach for the polynomial stability of a system of partially damped wave equations

In this paper, we study the stability of a system of wave equations which are weakly coupled and partially damped. Using a frequency domain approach based on the growth of the resolvent on the imaginary axis, we establish the polynomial energy decay rate for smooth initial data. We show that the behavior of the system is sensitive to the arithmetic property of the ratio of the wave propagation speeds of the two equations.     

On the interaction of deep water waves and exponential shear currents

A train of periodic deep-water waves propagating on a steady shear current with a vertical distribution of vorticity is investigated by an analytic method, namely the homotopy analysis method (HAM). The magnitude of the vorticity varies exponentially with the magnitude of the stream function, while remaining constant on a particular streamline. The so-called Dubreil–Jacotin transformation is used to transfer the original exponentially nonlinear boundary-value problem in an unknown domain into an algebraically nonlinear boundary-value problem in a known domain. Convergent series solutions are obtained not only for small amplitude water waves on a weak current but also for large amplitude waves on a strong current. The nonlinear wave-current interaction is studied in detail. It is found that an aiding shear current tends to enlarge the wave phase speed, sharpen the wave crest, but shorten the maximum wave height, while an opposing shear current has the opposite effect. Besides, the amplitude of waves and fluid velocity decay over the depth more quickly on an aiding shear current but more slowly on an opposing shear current than that of waves on still water. Furthermore, it is found that Stokes criteria of wave breaking is still valid for waves on a shear current: a train of propagating waves on a shear current breaks as the fiuid velocity at crest equals the wave phase speed. Especially, it is found that the highest waves on an opposing shear current are even higher and steeper than that of waves on still water. Mathematically, this analytic method is rather general in principle and can be employed to solve many types of nonlinear partial differential equations with variable coefficients in science, finance and engineering.     

微极热弹性无限板的轴对称自由振动

研究了在应力自由和刚性固定边界条件下,无能量耗散的均匀、各向同性微极热弹性无限板的轴对称自由振动波的传播,导出了相应的对称和斜对称模态波传播的闭合式特征方程和不同区域的特征方程.对短波的情况,应力自由热绝缘和等温板中对称和斜对称模态波传播的特征方程退化为Rayleigh表面波频率方程.根据导出的特征方程得到了热弹性、微极弹性和弹性板的结果.在对称和斜对称运动中计算了板的位移分量幅值、微转动幅值和温度分布,给出了对称和斜对称模式的频散曲线,并示出了位移分量和微转动幅值和温度分布的曲线.能够发现理论分析和数值结论是非常一致的.     

Wave localization and conversion phenomena in multi-coupled multi-span beams

The linear dynamics of nearly periodic disordered multi-span beams resting on flexible supports are investigated. A wave transfer matrix methodology is chosen to examine the propagation of waves and the transmission of vibration along the structure. The spans are bi-coupled through the rotation and the transverse displacement at the supports and thus the beam motion is made up of two independent wave types. While for the ordered infinite beam there exists frequency passbands for which the free harmonic waves propagate without attenuation, the introduction of a slight disorder among the span lengths results in the localization of the vibration energy to few spans and in the conversion of the energy from one type of wave to the other. The energy conversion phenomenon renders the mechanism of localization much more complex than in mono-coupled periodic systems. The contribution of each type of wave to the global beam motion is analyzed in terms of frequency. It is observed that the spatial decay of each wave type is mainly governed by an exponential envelope. The corresponding exponential decay constants define a measure of localization for each wave and are found to be equal to the Lyapunov exponents of the product of random wave transfer matrices. It is also found that at frequencies which belong to a passband for both wave types, the decay rate of an incident wave vector is bounded by the two Lyapunov exponents, while at frequencies which belong to a passband for one wave type and a stopband for the other, localization effects are best predicted by the smallest of the two Lyapunov exponents.     

Decay rates of energy of the 1D damped original nonlinear wave equation

We consider the decay rate of energy of the 1D damped original nonlinear wave equation. We first construct a new energy function. Then, employing the perturbed energy method and the generalized Young’s inequality, we prove that, with a general growth assumption on the nonlinear damping force near the origin, the decay rate of energy is governed by a dissipative ordinary differential equation. This allows us to recover the classical exponential, polynomial, or logarithmic decay rate for the linear, polynomial or exponentially degenerating damping force near the origin, respectively. Unlike the linear wave equation, the exponential decay rate constant depends on the initial data, due to the nonlinearity.