Riesz Basis Approach to the Stabilization of a Flexible Beam with a Tip Mass

Using an abstract condition of Riesz basis generation of discrete operators in the Hilbert spaces, we show, in this paper, that a sequence of generalized eigenfunctions of an Euler– Bernoulli beam equation with a tip mass under boundary linear feedback control forms a Riesz basis for the state Hilbert space. In the meanwhile, an asymptotic expression of eigenvalues and the exponential stability are readily obtained. The main results of [SIAM J. Control Optim., 36 (1998), pp. 1962–1986] are concluded as a special case, and the additional conditions imposed there are removed.