The influence of the physical coefficients of a Bresse system with one singular local viscous damping in the longitudinal displacement on its stabilization

Abstract

<p style='text-indent:20px;'>In this paper, we investigate the stabilization of a linear Bresse system with one singular local frictional damping acting in the longitudinal displacement, under fully Dirichlet boundary conditions. First, we prove the strong stability of our system. Next, using a frequency domain approach combined with the multiplier method, we establish the exponential stability of the solution if the three waves have the same speed of propagation. On the contrary, we prove that the energy of our system decays polynomially with rates <inline-formula><tex-math id="M1">\begin{document}$ t^{-1} $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M2">\begin{document}$ t^{-\frac{1}{2}} $\end{document}</tex-math></inline-formula>.</p>

Stabilization of the generalized Rao‐Nakra beam by partial viscous damping

“…For a system of partially damped wave equations coupled by velocity or displacement (such as the Timoshenko beam equation, the Bresse beam equation), it is well‐known that the damping is more effective in the case of equal wave speeds (see previous works 20–23 ). However, for the Rao‐Nakra sandwich beam equation or its generalized version, the opposite is true.…”

Stability results of locally coupled wave equations with local Kelvin-Voigt damping: Cases when the supports of damping and coupling coefficients are disjoint

“…The direct and indirect stability of locally coupled wave equations with local damping arouses many interests in recent years. The study of coupled systems is also motivated by several physical considerations like Timoshenko and Bresse systems (see for instance [10,6,3,2,1,15,14]). The exponential or polynomial stability of the wave equation with a local Kelvin-Voigt damping is considered in [20,23,13], for instance.…”