Asymptotic stabilization for Bresse transmission systems with fractional damping

Abstract

In this article, we study the asymptotic stability of Bresse transmission systems with two fractional dampings. The dissipation mechanism of control is given by the fractional damping term and acts on two equations. The relationship between the stability of the system, the fractional damping index \(\theta\in[0,1]\) and the different wave velocities is obtained. By using the semigroup method, we obtain the well-posedness of the system. We also prove that when the wave velocities are unequal or equal with \(\theta\neq 0\), the system is not exponential stable, and it is polynomial stable. In addition, the precise decay rate is obtained by the multiplier method and the frequency domain method. When the wave velocities are equal with \(\theta=0\), the system is exponential stable. For more information see https://ejde.math.txstate.edu/Volumes/2023/87/abstr.html