Spectral analysis and stabilization of a coupled wave-ODE system

In this paper, an interconnected wave-ODE system with K-V damping in the wave equation and unknown parameters in the ODE is considered. It is found that the spectrum of the system operator is composed of two parts: Point spectrum and continuous spectrum. The continuous spectrum consists of an isolated point $$- \tfrac{1}{d}$$, and there are two branches of the asymptotic eigenvalues: The first branch is accumulating towards $$- \tfrac{1}{d}$$, and the other branch tends to −∞. It is shown that there is a sequence of generalized eigenfunctions, which forms a Riesz basis for the Hilbert state space. As a consequence, the spectrum-determined growth condition and exponential stability of the system are concluded.