Stability and Spectral Properties of General Tree-Shaped Wave Networks with Variable Coefficients

Abstract

The stability of general tree-shaped wave networks with variable coefficients under boundary feedback controls is considered. Making full use of the tree-shaped structures, we present a detailed asymptotic spectral analysis of the networks. By proposing the from-root-to-leaf calculating technique, we deduce an explicit recursive expression for the asymptotic characteristic equation and the spectral properties are further obtained. We show that the spectrum-determined-growth (SDG) condition holds. Thus the stability analysis of the closed-loop system can be completely converted to the infimum estimation of the asymptotic characteristic equation. Especially, we further show that the infimum is positive so as to obtain the exponential stability by estimating the recursive expression in from-leaf-to-root order. Some numerical simulations are presented to illustrate and support the theoretical results.