In this paper, we consider stability for a hyperbolic-hyperbolic coupled system consisting of Euler-Bernoulli beam and wave equations, where the structural damping of the wave equation is taken into account. The coupling is actuated through boundary weak connection in the sense that after differentiation of the total energy for coupled system, only the term of the wave equation appears explicitly. We first show that the spectrum of the closed-loop system consists of three branches: one branch is basically along the real axis and accumulates to a finite point; the second branch is also along the real line; and the third branch distributes along two parabola likewise symmetric with the real axis. The asymptotic expressions of both eigenvalues and eigenfunctions are obtained by means of asymptotic analysis. With an estimation of the resolvent operator, the completeness of the root subspace is proved. The Riesz basis property and exponential stability of the system are then concluded. Finally, we show that the associated C_0-semigroup is of Gevrey class, which shows that not only the stability but also regularity have been transmitted from regular wave subsystem to the whole system through this boundary connections.

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In this paper, we study stability for a Schrödinger equation interacted by an Euler-Bernoulli beam equation with Kelvin-Voigt damping through weak boundary connections. It is shown that the whole coupled system is well-posed. With a careful spectral analysis, it is shown that the system operator of the closed-loop system is not of compact resolvent and the spectrum consists of three branches. By means of asymptotic analysis, the asymptotic expressions of eigenfunctions are obtained. The Riesz basis property and exponential stability of the system are concluded by comparison method in Riesz basis approach. Finally, we show that the associated 0 -semigroup is of Gevery class > 4 which is a remarkable difference with the related literature.