The stability number of the Timoshenko system with second sound

In this paper we consider the following Timoshenko system: in (0, L) × ℝ + {ρ 1 ϕ tt -k 1 (ϕ x +ψ) x =0 ρ 2 ψ tt - k 2 ψ xx + ∫ t 0 g(t - τ)(a(x)ψ x (τ)) x dτ+k 1 (ϕ x + ψ) + b(x)h(ψ t ) =0 in (0, L) × ℝ + with Dirichlet boundary conditions and initial data where a, b, g and h are specific functions and ρ 1 , ρ 2 , κ 1 , κ 2 and L are given positive constants. We establish a general stability estimate using the multiplier method and some properties of convex functions. Without imposing any growth condition on h at the origin, we show that the energy of the system is bounded above by a quantity, depending on g and h, which tends to zero as time goes to infinity. Our estimate allows us to consider a large class of functions h with general growth at the origin. We use some examples (known in the case of wave equations and Maxwell system) to show how to derive from our general estimate the polynomial, exponential or logarithmic decay. The results of this paper improve and generalize some existing results in the literature and generate some interesting open problems.

General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping

Energy decay in a Timoshenko-type system of thermoelasticity of type III

In this paper we consider the following Timoshenko system
\begin{eqnarray*}
\varphi _{t t}-(\varphi _{x}+\psi )_{x}=0,\quad (0,1)\times 
R^+\\
\psi _{t t}-\psi _{x x}+\varphi _{x}+\psi +\int_0^t g(t-\tau )\psi_{x x}(\tau )d\tau =0,\quad (0,1)\times R^{+}
\end{eqnarray*}
with Dirichlet boundary conditions where $g$ is a positive nonincreasing
function satisfying
\begin{eqnarray*}
g'(t)\leq -H(g(t))
\end{eqnarray*}
and $H$ is a function satisfying some regularity and convexity conditions.
We establish a general stability result for this system.

A general stability result in a memory-type Timoshenko system

In this paper, we consider nonlinear thermoelastic systems of Timoshenko type in a one-dimensional bounded domain. The system has two dissipative mechanisms being present in the equation for transverse displacement and rotation angle—a frictional damping and a dissipation through hyperbolic heat conduction modelled by Cattaneo's law, respectively. The global existence of small, smooth solutions and the exponential stability in linear and nonlinear cases are established. Copyright © 2008 John Wiley & Sons, Ltd.

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Nonlinear damped Timoshenko systems with second sound—Global existence and exponential stability

We study the uniform stabilization of a Timoshenko beam by one control force. We prove that under, one locally distributed damping, the exponential stability for this system is assured if and only if the wave speeds are the same.

Ali Wehbe

Papers

Uniform stabilization for the timoshenko beam by a locally distributed damping