The spectral analysis and exponential stability of a bi-directional coupled wave-ODE system

In this paper, an unstable linear time invariant (LTI) ODE system X(t) = AX(t) is stabilized exponentially by the PDE compensato—a wave equation with Kelvin-Voigt (K-V) damping. Direct feedback connections between the ODE system and wave equation are established: The velocity of the wave equation enters the ODE through the variable v _t (1,t); meanwhile, the output of the ODE is fluxed into the wave equation. 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 −1/d, and there are two branches of asymptotic eigenvalues: the first branch approaches to −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.