# Stability of elastic transmission systems with a local Kelvin-Voigt damping

## TL;DR

This study analyzes the stability of elastic transmission beams with localized Kelvin-Voigt damping, showing exponential stability for transversal vibrations and polynomial stability for longitudinal vibrations using advanced mathematical techniques.

## Contribution

It introduces a novel analysis of stability for transmission Euler-Bernoulli beams with local Kelvin-Voigt damping, combining frequency domain and multiplier methods.

## Key findings

- Transversal vibrations are exponentially stable.
- Longitudinal vibrations are polynomially stable.
- The damping's local and unbounded nature requires specialized analysis.

## Abstract

In this paper, we consider the longitudinal and transversal vibrations of the transmission Euler-Bernoulli beam with Kelvin-Voigt damping distributed locally on any subinterval of the region occupied by the beam and only in one side of the transmission point. We prove that the semigroup associated with the equation for the transversal motion of the beam is exponentially stable, although the semigroup associated with the equation for the longitudinal motion of the beam is polynomially stable. Due to the locally distributed and unbounded nature of the damping, we use a frequency domain method and combine a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent.