Asymptotic stability and Riesz basis property for tree-shaped network of strings

Abstract

This paper discusses the asymptotic stability and Riesz basis generation for a general tree-shaped network of vibrating strings. All exterior vertices are assumed to be fixed and interior vertices are imposed linear damping feedbacks. This paper shows that the system is well-posed and asymptotically stable by C0-semigroup theory. With some additional conditions, the spectrum of the system is shown to be located in a strip that is parallel to the imaginary axis and the set of all generalized eigenfunctions is completed in the state space. These lead to the conclusion that there is a sequence of generalized eigenfunctions of the system, which forms a Riesz basis with parenthesis for the state space.