Riesz basis of exponential family for a hyperbolic system

Abstract

This paper studies a linear hyperbolic system with boundary conditions that was first studied under some weaker conditions in [8, 11]. Problems on the expansion of a semigroup and a criterion for being a Riesz basis are discussed in the present paper. It is shown that the associated linear system is the infinitesimal generator of a {C_{0}} -semigroup; its spectrum consists of zeros of a sine-type function, and its exponential system {\{e^{\lambda_{n}t}\}_{n\geq 1}} constitutes a Riesz basis in {L^{2}[0,T]} . Furthermore, by the spectral analysis method, it is also shown that the linear system has a sequence of eigenvectors, which form a Riesz basis in Hilbert space, and hence the spectrum-determined growth condition is deduced.