Basis property and stabilization of a translating tensioned beam through a pointwise control force

✍ Scribed by Bao-Zhu Guo; Yu Xie

• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 701 KB
• Volume: 47
• Category: Article
• ISSN: 0898-1221
• DOI: 10.1016/s0898-1221(04)90132-0

✦ Synopsis

Introducing a recently developed powerful method, Riesz basis method as it is called, and applying it to the stabilization of a translating tensioned beam through a pointwise control force are the two purposes of the paper. An asymptotic analysis, which is different from the one in treating the same problem in literature, is used to get a complete asymptotic expansion of the eigenvaiues and eigenfunctions for the beam. The controlled system is written as an evolution equation in a state Hilbert space whose energy nbrm (induced by inner product) is the total energy of the beam. The spectrum-determined growth condition is obtained by showing that there is a set of generalized eigenfunctions of the system, which forms a Riesz basis for the state space. As a consequence, the stability, particularly the exponential stability of the system, is developed. The analysis used here can be extended to other distributed systems and the results significantly improve that in [1]. Q 2004 Elsevier Ltd. All rights reserved.