On the dynamic behavior and stability of controlled connected Rayleigh beams under pointwise output feedback

¹ Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100080, P.R. China; bzguo@iss.ac.cn
² School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa.
³ Corresponding author: Department of Mathematics, Beijing Institute of Technology, Beijing 100081, P.R. China; wangjc@graduate.hku.hk
⁴ Department of Mathematics, Beijing Institute of Technology, Beijing 100081, P.R. China.

Received: 12 April 2006
Revised: 23 October 2006
Revised: 16 March 2007

Abstract

We study the dynamic behavior and stability of two connected Rayleigh beams that are subject to, in addition to two sensors and two actuators applied at the joint point, one of the actuators also specially distributed along the beams. We show that with the distributed control employed, there is a set of generalized eigenfunctions of the closed-loop system, which forms a Riesz basis with parenthesis for the state space. Then both the spectrum-determined growth condition and exponential stability are concluded for the system. Moreover, we show that the exponential stability is independent of the location of the joint. The range of the feedback gains that guarantee the system to be exponentially stable is identified.