Riesz basis property, exponential stability of variable coefficient Euler–Bernoulli beams with indefinite damping

Abstract

We study damped Euler–Bernoulli beams that have nonuniform thickness or density. These nonuniformfeatures result in variable coefficient beam equations. We prove that despite the nonuniform features, the eigenfunctions of the beam form a Riesz basis and asymptotic behaviour of the beam system can be deduced without any restrictions on the sign of the damping. We also provide an answer to the frequently asked question on damping: ‘how much more positive than negative should the damping be without disrupting the exponential stability?’, and result in a criterion condition which ensures that the system is exponentially stable.