Riesz Basis Property and Stability of Planar Networks of Controlled Strings

In this paper we study the basis property of root vectors of a star-shaped networks of strings whose exterior ends are clamped and common node has a damping. The main operator determined by the networks is non-normal. By the asymptotical technique, we show that its spectra distribute in a strip parallel to the imaginary axis, and there is a sequence of root vectors (generalized eigenvectors and eigenvectors) that forms a Riesz basis with parentheses for the Hilbert state space. As a application of this result, we discuss the stability of the system.