Spectral asymptotics, instability and Riesz basis property of root vectors for Rayleigh beam model with non-dissipative boundary conditions

Abstract

In this paper we investigate the Rayleigh beam model with non-dissipative boundary conditions recently considered in the literature. The beam is clamped at the left end and subject to a feedback control type boundary conditions at the right. The components of the 2-dim input vector are shear and moment at the right end and the components of the observation vector are time derivatives of displacement and slope at the right end. The input is related to the observation through a co-diagonal matrix depending on two non-negative control parameters. The paper contains two main results. First we derive the leading term, the second order term, and the remainder in the asymptotic representation of the eigenmodes of the system. It follows from spectral asymptotics that the system has an infinite alternating sequence of stable and unstable eigenmodes. This type of instability has been recently obtained in the literature by a different method. The instability result might be of interest for energy harvesting models where destabilization of a system is desirable. The second result is the fact that the generalized eigenvectors of the dynamics generator of the system form a Riesz basis in the state space equipped with the energy metric. This result implies, in particular, that the system generates a C0-semigroup in the state space.