Riesz Basis Property of Timoshenko Beams with Boundary Feedback Control

A Timoshenko beam equation with boundary feedback control is considered. By an abstract result on the Riesz basis generation for the discrete operators in the Hilbert spaces, we show that the closed-loop system is a Riesz system, that is, the sequence of generalized eigenvectors of the closed-loop system forms a Riesz basis in the state Hilbert space. 1. Introduction. The boundary feedback stabilization problem of a hybrid system has been studied extensively in the last decade. Many important results have been obtained. Among them, most of studies in the literatures are concerned with Euler-Bernoulli and Rayleigh beams; there are a few results for Timoshenko beams (cf. [3, 5, 6, 7, 9]), which are mainly focused on the stability of the closed-loop system. Though it is important to obtain the exponential stability of the system, it is also very interesting to study the rate of the exponential decay of the system. It is well known that if the system satisfies the spectrum-determined growth assumption, then the rate of the exponential decay can be easily estimated via the spectra of the system operator, see [2]. Furthermore, if the system is a Riesz one, that is, the set of the generalized eigenvectors of the system operator forms a Riesz basis of the state Hilbert space, then the spectrum-determined growth assumption is satisfied. In [1], the Riesz basis property was used to give some quantitative information of the rate of the exponential decay for a simpler Euler-Bernoulli beam system with no tip mass. For Euler-Bernoulli and Rayleigh beam systems, some further results concerning the Riesz basis property of the systems can be found in [4, 8]. In the present note, we consider the following Timoshenko beam equation with a tip mass (see [7, 9]):