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

Guo, Bao-Zhu; Ren, Han-Jing

ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 73.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

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.

Classification : 47B06, 74K10, 93D15, 70J25
Mots-clés : Euler-Bernoulli beam equation, wave equation with structural damping, coupled system, Gevrey class

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