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

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.