Stability results of a distributed problem involving Bresse system with history and/or Cattaneo law under fully Dirichlet or mixed boundary conditions

TL;DR

This paper investigates the stability of a one-dimensional Bresse system with memory and heat conduction effects under various boundary conditions, establishing exponential stability in some cases and polynomial decay in others.

Contribution

It extends previous stability analyses by considering fully Dirichlet boundary conditions and different propagation speeds, providing new decay rate results for the Bresse system.

Findings

Exponential stability under equal propagation speeds.

Polynomial decay when propagation speeds differ.

Generalization to mixed boundary conditions.

Abstract

In this paper, we study the stability of a one-dimensional Bresse system with infinite memory type control and/or with heat conduction given by Cattaneo's law acting in the shear angle displacement. When the thermal effect vanishes, the system becomes elastic with memory term acting on one equation. Unlike [6], [10], and [22], we consider the interesting case of fully Dirichlet boundary conditions. Indeed, under equal speed of propagation condition, we establish the exponential stability of the system. However, in the natural physical case when the speeds of propagation are different, using a spectrum method, we show that the Bresse system is not uniformly stable. In this case, we establish a polynomial energy decay rate. Our study is valid for all other mixed boundary conditions and generalizes that of [6], [10], and [22].