Riesz Basis Property for Generic Network of Strings

In this paper, we study the generation problem of Riesz basis for a generic network of strings with joint damping at all vertices. First we give spectral basic property of system operator A. Under certain conditions we prove that the spectrum of A distributes in a strip parallel to the imaginary axis. By the discussion of completeness of the generalized eigenvectors of operator A, we prove further that there exists a sequence of generalized eigenvectors of A that forms a Riesz basis with parentheses in the state Hilbert space.