Spectrum and dynamical behavior of a kind of planar network of non-uniform strings with non-collocated feedbacks

A kind of planar network of strings with non-collocated terms in boundary feedback controls is considered. Suppose that the network is constituted by $n$ non-uniform strings, connected by one vibrating point mass. The displacements of these strings are continuous at the common vertex. The non-collocated terms are contained in feedback controls at exterior nodes. The well-posedness of the corresponding closed-loop system is proved. A complete spectral analysis is carried out and the asymptotic expression of the spectrum of this system operator is obtained, which implies that the asymptotic behavior of the spectrum is independent of these non-collocated terms. Then the Riesz basis property of the (generalized) eigenvectors of the system operator is proved. Thus, the spectrum determined growth condition holds. Finally, the exponential stability of a special case of this kind of network is gotten under certain conditions. In order to support these results, a numerical simulation is given.