Riesz basis and stabilization for the flexible structure of a symmetric tree-shaped beam network

The stabilization of a symmetric tree-shaped network of Euler-Bernoulli beams described by a system of partial differential equations is considered. The boundary controllers are designed based on passivity principle. The eigenfrequencies are analysed in detail and the asymptotic expansion of eigenvalues are presented. It is shown that there is a set of generalized eigenfunctions for the closed-loop system, which forms a Riesz basis with parentheses for the energy state space. This concludes the spectrum-determined growth condition and the exponential stability of the closed-loop system.