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

Abstract

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 eigenvalues and eigenfunctions for the beam. The controlled system is written as an evolution equation in a state Hilbert space whose energy norm (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].