Riesz basis property of serially connected Timoshenko beams

Abstract

In this paper we study the Riesz basis property of serially connected Timoshenko beams with joint and boundary feedback controls. Suppose that the left end of the whole beam is clamped and the right end is free. At intermediate nodes, the displacement and rotational angle of beams are continuous but the shearing force and bending moment could be discontinuous. The collocated velocity feedback of the beams at intermediate nodes and the right end are used to stabilize the system. We prove that the operator determined by the closed loop system has compact resolvent and generates a C ₀ semigroup in an appropriate Hilbert space. We also show that there is a sequence of the generalized eigenvectors of the operator that forms a Riesz basis with parentheses. Hence the spectrum determined growth condition holds. Therefore if the imaginary axis is not an asymptote of the spectrum, then the closed loop system is exponentially stable. Finally, we give a conclusion remark to explain that our result can be applied not only on the serially connected Timoshenko beams.

Acknowledgement

The authors would like to thank the referees for their very constructive and useful suggestions and comments. This research is supported by the Natural Science Foundation of China grant NGFC-60474017 695 and by the HKRGC Grants of code 7059/06P and the HKUCRCG Grant of code 10206052.