The lack of exponential stability of a Bresse system subjected only to two dampings

TL;DR

This paper investigates the stability properties of a Bresse system with two dampings, revealing that exponential stability does not occur due to eigenvalue asymptotics, especially in the case of three distinct wave speeds.

Contribution

It provides a mathematical analysis of the eigenvalues and eigenvectors of a Bresse system with limited damping, showing the absence of exponential stability in certain cases.

Findings

Eigenvalues asymptote to the imaginary axis

Exponential stability is not achieved

Analysis focused on three distinct wave speeds

Abstract

In this paper, we study the indirect boundary stabilization of a Bresse system with only two dissipation laws. This system, which models the dynamics of a beam, is a hyperbolic system with three wave speeds. We study the asymptotic behaviour of the eigenvalues and of the eigenvectors of the underlying operator in the case of three distinct wave velocities which is not physically relevant. Since the imaginary axis is proved to be an asymptote for one family of eigenvalues, the stability can not be exponential. Of course, this paper is only interesting from a mathematical point of view.