On the completeness of root vectors generated by systems of coupled hyperbolic equations

Abstract

The paper is the second in a set of two papers, which are devoted to a unified approach to the problem of completeness of the generalized eigenvectors (the root vectors) for a specific class of linear non‐selfadjoint unbounded matrix differential operators. The list of the problems for which such operators are the dynamics generators includes the following: (a) initial boundary‐value problem (IBVP) for a non‐homogeneous string with both distributed and boundary damping; (b) IBVP for small vibrations of an ideal filament with a one‐parameter family of dissipative boundary conditions at one end and with a heavy load at the other end; this filament problem is treated for two cases of the boundary parameter: non‐singular and singular; (c) IBVP for a three‐dimensional damped wave equation with spherically symmetric coefficients and both distributed and boundary damping; (d) IBVP for a system of two coupled hyperbolic equations constituting a Timoshenko beam model with variable coefficients and boundary damping; (e) IBVP for a coupled Euler‐Bernoulli and Timoshenko beam model with boundary energy dissipation (the model known in engineering literature as bending‐torsion vibration model); (f) IBVP for two coupled Timoshenko beams model, which is currently accepted as an appropriate model describing vibrational behavior of a longer double‐walled carbon nanotube. Problems (a)–(c) have been discussed in the first paper of the aforementioned set. Problems (d)–(f) are discussed in the present paper.

On completeness of the system of eigenfunctions of the Dirichlet problem for Lame equations in the strip

“…The inverse statements also hold. Using any solution to the system of equations (5), corresponding to a simple eigenvalue of the matrix M, it is easy to construct a solution to the BVP.…”

“…The completeness of generalized eigenfunctions of a polynomial nonselfadjoint operator pencil has been addressed in [4]. The question of the completeness of root vectors generated by systems of coupled hyperbolic equations is considered in [5].…”