Riesz basis generation of a serially connected string system under joint damping feedbacks

An abstract sufficient condition is developed to deal with Riesz basis generation in Hilbert spaces for cases where the eigenvalues are not necessarily simple and separable but are comprised of some finite unification of separable sets. The condition is then applied to the connected string system to show that there is a family of generalized eigenfunctions, which forms a Riesz basis with parentheses in the state space. The spectrum-determined growth condition is concluded as a consequence.