Stability of an $N$-component Timoshenko beam with localized Kelvin-Voigt and frictional dissipation

Abstract

We consider the transmission problem of a Timoshenko's beam composed by N components, each of them being either purely elastic, or a Kelvin-Voigt viscoelastic material, or an elastic material inserted with a frictional damping mechanism. Our main result is that the rate of decay depends on the position of each component. More precisely, we prove that the Timoshenko's model is exponentially stable if and only if all the elastic components are connected with one component with frictional damping. Otherwise, there is no exponential stability, but a polynomial decay of the energy as $1/t^2$. We introduce a new criterion to show the lack of exponential stability, Theorem 1.2. We also consider the semilinear problem.

Keywords

• Timoshenko's model
• beam equation
• localized dissipation
• viscoelaticity
• lack of exponential stability
• exponential and polynomial stability