Uniform Stabilization of a Hybrid System of Elasticity: Riesz Basis Approach

A hybrid system, composed of an elastic beam governed by an Euler-Bernoulli beam equation and a linked rigid body governed by an ordinary differential equation, is considered. This paper studies the basis property and the stability of a hybrid system when the usual linear boundary feedback is applied to the end without mass. It is shown that there is a sequence of generalized eigenfunctions of the system, which forms a Riesz basis for the state Hilbert space. As consequence expressions of eigenvalues, the spectrum-determined growth condition and the exponential stability are readily presented. To confirm numerically the asymptotic estimate of eigenvalues, we shall use the spectral method to calculate the eigenvalues.