Stability and Riesz basis property for general network of strings

Abstract

In this paper, we study the generation problem of Riesz basis for a general network of strings with joint damping at each vertex. First, we give a basic spectral property of the system operator $$ \mathcal{A} $$ . Under certain conditions, we prove that the spectrum of $$ \mathcal{A} $$ is distributed in a strip parallel to the imaginary axis. By the discussion of the completeness of generalized eigenvectors of the operator $$ \mathcal{A} $$ , we prove further that there exists a sequence of generalized eigenvectors of $$ \mathcal{A} $$ that forms a Riesz basis with parentheses in the Hilbert state space.