Generation of Gevrey class semigroup by non-selfadjoint Euler-Bernoulli beam model

• Published in: Mathematical methods in the applied sciences 2006-12, Vol.29 (18), p.2181-2199
• Main Author: Shubov, Marianna A.
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; Fundamental areas of phenomenology (including applications); Gevrey class semigroup; Mathematical analysis; Mathematics; Partial differential equations; Physics; Riesz basis property; Sciences and techniques of general use; Solid mechanics; Structural and continuum mechanics; Structural mechanics (beam, string...); Theory and numerical methods
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.768

Asymptotic and spectral properties of a non‐selfadjoint operator that is a dynamics generator for the Euler–Bernoulli beam model of a finite length are studied in this paper. The hyperbolic equation, which governs the vibrations of the Euler–Bernoulli beam model, is supplied with a one‐parameter family of physically meaningful boundary conditions containing damping terms. The initial boundary‐value problem is equivalent to the evolution equation that generates a strongly continuous semigroup in the state space of the system. It is found that the semigroup, being non‐analytic, belongs to Gevrey class semigroups. This means that the differentiability of such semigroup is slightly weaker than that of an analytic semigroup. In the forthcoming works, the results of the present paper will be applied (a) to the solution of the exact controllability problem for Euler–Bernoulli beam and (b) to spectral analysis of a planar network of serially connected Euler–Bernoulli beams modelling ‘flying wing configurations’ in aeronautic engineering. Copyright © 2006 John Wiley & Sons, Ltd.