Stability and regularity transmission for coupled beam and wave equations through boundary weak connections^*

¹ Key Laboratory of System and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P.R. China.
² School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, P.R. China.
³ School of Mathematics and Physics, North China Electric Power University, Beijing 102206, P.R. China.

^** Corresponding author: bzguo@iss.ac.cn

Received: 26 August 2018
Accepted: 5 September 2019

Abstract

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 C₀-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.