Spectral operators generated by Timoshenko beam model

Abstract

We announce a series of results on the spectral analysis for a class of nonselfadjoint operators which are the dynamics generators for the systems governed by the equations of the Timoshenko beam model with a 2-parameter family of dissipative boundary conditions. Our results split into three groups. (1) We present asymptotic formulas for the spectra of the aforementioned operators (the spectrum of each operator consists of two branches of discrete complex eigenvalues) and for their generalized eigenvectors. (2) We show that these operators are Riesz spectral. This result follows from the fact that the systems of generalized eigenvectors form Riesz bases in the corresponding energy spaces. (3) We give the asymptotics of the spectra and the eigenfunctions for the nonselfadjoint polynomial operator pencils associated with these operators. Our results, on one hand, provide a class of nontrivial examples of spectral operators (nonselfadjoint operators which admit an analog of spectral decomposition). On the other hand, these results give a key to the solutions of various control and stabilization problems for the Timoshenko beam model using the spectral decomposition method.