Dynamic boundary stabilization of Euler-Bernoulli beam through a Kelvin-Voigt damped wave equation

In this paper, we study the stability of a one-dimentional Euler-Bernoulli beam coupled with a Kelvin-Voigt damped wave equation, where the wave equation acts as a dynamic boundary feedback controller to exponentially stabilize the Euler-Bernoulli beam. Remarkably, the resolvent of the closed-loop system operator is not compact anymore. By a detailed spectral analysis, we show that the residual spectrum is empty and the continuous spectrum contains only one point. Moreover, we verify that the generalized eigenfunctions of the system forms a Riesz basis for the energy state space. It then follows that the C₀-semigroup generated by the system operator satisfies the spectrum-determined growth assumption. Finally, the exponential stability and Gevrey regularity of the system are established.