Exponential stability of Rayleigh beam equation on a star‐shaped network with indefinite damping

摘要

In this paper, we study the stability of the Rayleigh beam equation on a star‐shaped network with indefinite sign damping terms. We prove that the system is well posed via the semigroup theory. Then, by a detailed spectral analysis of the system operator, we provide a necessary condition for the distribution of the spectrum, the completeness and the Riesz basis property of the eigenfunctions of the operator as well as the asymptotic stability of the system. Moreover, we discuss how the stability of the system depends on the sign of damping terms. More precisely, we obtain a condition on the negativity degree of the damping coefficients in order that the system always remains exponentially stable.