Energy Decay of some boundary coupled systems involving wave$\backslash$ Euler-Bernoulli beam with one locally singular fractional Kelvin-Voigt damping

TL;DR

This paper studies how energy diminishes over time in coupled hyperbolic systems involving wave and Euler-Bernoulli beam equations with a localized fractional Kelvin-Voigt damping, proving strong stability and polynomial decay rates.

Contribution

It introduces a novel analysis of energy decay in coupled systems with localized fractional damping, using augmented models and frequency domain techniques.

Findings

Models are strongly stable under certain conditions.

Energy decay rates are polynomial and depend on fractional order.

Decay rates vary with the type of damped equation.

Abstract

In this paper, we investigate the energy decay of hyperbolic systems of wave-wave, wave-Euler- Bernoulli beam and beam-beam types. The two equations are coupled through boundary connection with only one localized non-smooth fractional Kelvin-Voigt damping. First, we reformulate each system into an augmented model and using a general criteria of Arendt-Batty, we prove that our models are strongly stable. Next, by using frequency domain approach, combined with multiplier technique and some interpolation inequalities, we establish different types of polynomial energy decay rate which depends on the order of the fractional derivative and the type of the damped equation in the system.