Optimal indirect stability of a weakly damped elastic abstract system of second order equations coupled by velocities

TL;DR

This paper investigates the stability of a coupled second order elastic system with weak damping, demonstrating exponential stability under certain conditions and optimal polynomial decay otherwise, using the Riesz basis approach.

Contribution

It introduces a novel analysis of stability for coupled second order systems with fractional damping, establishing conditions for exponential and polynomial decay rates.

Findings

Exponential stability when damping is viscous and wave speeds are equal.

Optimal polynomial decay rate in other cases.

Illustrative examples demonstrating theoretical results.

Abstract

In this paper, by means of the Riesz basis approach, we study the stability of a weakly damped system of two second order evolution equations coupled through the velocities. If the fractional order damping becomes viscous and the waves propagate with equal speeds, we prove exponential stability of the system and, otherwise, we establish an optimal polynomial decay rate. Finally, we provide some illustrative examples.