Stability of the Timoshenko Beam Equation with One Weakly Degenerate Local Kelvin-Voigt Damping

TL;DR

This paper proves that a Timoshenko beam with localized Kelvin-Voigt damping, even with singularities, exhibits polynomial energy decay at a rate of t^{-1/2}.

Contribution

It establishes polynomial stability for the Timoshenko beam with weakly degenerate local Kelvin-Voigt damping using frequency domain and multiplier techniques.

Findings

The system decays polynomially at rate t^{-1/2}.

Stability holds regardless of damping acting on shear stress or bending moment.

The damping coefficient's singularity does not prevent polynomial decay.

Abstract

We consider the Timoshenko beam equation with locally distributed Kelvin-Voigt damping, which affects either the shear stress or the bending moment. The damping coefficient exhibits a singularity, causing its derivative to be discontinuous. By using the frequency domain method and multiplier technique, we prove that the associated semigroup is polynomial stability. Specifically, regardless of whether the local Kelvin-Voigt damping acts on the shear stress or the bending moment, the system decays polynomially with rate $t^{-\frac{1}{2}}$.