Riesz Basis Property of Evolution Equations in Hilbert Spaces and Application to a Coupled String Equation

Abstract

Suppose that $\{\lambda _n\}$ is the set of zeros of a sine-type generating function of the exponential system $\{e^{i\lambda_n t}\}$ in L2 (0,T) and is separated. Levin and Golovin's classical theorem claims that $\{e^{i\lambda_n t}\}$ forms a Riesz basis for L2 (0,T). In this article, we relate this result with Riesz basis generation of eigenvectors of the system operator of the linear time-invariant evolution equation in Hilbert spaces through its spectrum. A practically favorable necessary and sufficient condition for the separability of zeros of function of sine type is derived. The result is applied to get Riesz basis generation of a coupled string equation with joint dissipative feedback control.