Spectral operators generated by bending–torsion vibration model with two-end energy dissipation

Article type: Research Article

Authors: Shubov, Marianna A.

Affiliations: Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824, USA E-mail: marianna.shubov@euclid.unh.edu

Abstract: This research is devoted to the asymptotic and spectral analysis of a coupled Euler–Bernoulli and Timoshenko beam model. The model is governed by a system of two coupled hyperbolic partial differential equations and a four-parameter family of the boundary conditions modeling the action of self-straining actuators. The aforementioned system of equations of motion together with the boundary conditions is equivalent to a single operator evolution equation in the energy space. That equation defines a semigroup of bounded operators. The dynamics generator of the semigroup is our main object of interest. For each set of boundary parameters, the dynamics generator has a compact inverse. If all four boundary parameters are not purely imaginary numbers, then the dynamics generator is a nonself-adjoint operator in the energy space. We present two main results in this research. As the first one, we calculate the spectral asymptotics of the dynamics generator. We find that the complex spectrum lies in a strip parallel to the real axis, and is asymptotically close to the axis. The latter fact means that the system is stable, but is not uniformly stable. As the second main result, we prove that the set of the root vectors of the dynamics generator forms a Riesz basis in the energy space. The results obtained in the present paper allow us to conclude that the dynamics generator is a Riesz spectral operator in the sense of N. Dunford.

Keywords: matrix differential operator, spectrum, root vectors, completeness, minimality, Riesz basis property

Journal: Asymptotic Analysis, vol. 45, no. 1-2, pp. 133-169, 2005