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Asymptotic and spectral results on the initial boundary‐value problem for the coupled bending‐torsion vibration model (which is important in such areas of engineering sciences as bridge and tall building designs, aerospace and oil pipes modeling, etc.) are presented. The model is given by a system of two hyperbolic partial differential equations equipped with a three‐parameter family of non‐self‐adjoint (linear feedback type) boundary conditions modeling the actions of self‐straining actuators. The system is rewritten in the form of the first‐order evolution equation in a Hilbert space of a four‐component Cauchy data. It is shown that the dynamics generator is a matrix differential operator with compact resolvent, whose discrete spectrum splits asymptotically into two disjoint subsets called the α‐branch and the β‐branch, respectively. Precise spectral asymptotics for the eigenvalues from each branch as the number of an eigenvalue tends to ∞ have been derived. It is also shown that the leading asymptotical term of the α‐branch eigenvalue depends only on the torsion control parameter, while of the β‐branch eigenvalue depends on two bending control parameters. [ABSTRACT FROM AUTHOR]

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 [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a recently developed mathematical model for a short double-wall carbon nanotube. The model is governed by a system of two coupled hyperbolic equations and is reduced to an evolution equation. This equation defines a dissipative semi-group. We show that the semi-group generator is an unbounded nonselfadjoint operator with compact resolvent. Moreover, this operator is a relatively compact perturbation of a certain selfadjoint operator. [Copyright &y& Elsevier]

The zero controllability problem for the system of two coupled hyperbolic equations which governs the vibrations of the coupled Euler–Bernoulli and Timoshenko beam model is studied in the paper. The system is considered on a finite interval with a two-parameter family of physically meaningful boundary conditions containing damping terms. The controls are introduced as separable forcing terms gi(x)fi(t), i = 1, 2, on the right-hand sides of both equations. The force profile functions gi(x), i = 1, 2, are assumed to be given. To construct the controls fi(t), i = 1, 2, which bring a given initial state of the system to zero on the specific time interval [0, T], the spectral decomposition method has been applied. The approach, used in the present paper, is based on the results obtained in the recent works by the author and the collaborators. In these works, the detailed asymptotic and spectral analyses of the non-self-adjoint operators generating the dynamics of the coupled beam have been carried out. It has been shown that for each set of the boundary parameters, the aforementioned operator is Riesz spectral, i.e. its generalized eigenvectors form a Riesz basis in the energy space. Explicit asymptotic formulas for the two-branch spectrum have also been derived. Based on these spectral results, the control problem has been reduced to the corresponding moment problem. To solve this moment problem, the asymptotical representation of the spectrum and the Riesz basis property of the generalized eigenvectors have been used. The necessary and/or sufficient conditions for the exact controllability are proven in the paper and the explicit formulas for the control laws are given. The case of the approximate controllability is discussed in the paper as well. [ABSTRACT FROM PUBLISHER]

The present paper is devoted to the asymptotic and spectral analysis of a system of coupled Euler-Bernoulli and Timoshenko beams. The model is governed by a system of two coupled differential equations and a two parameter family of boundary conditions modelling the action of the self-straining actuators. The above equations of motion form a coupled linear hyperbolic system, which is equivalent to a single operator evolution equation in the energy space. That equation defines a semigroup of bounded operators. This is a dynamics generator of the semigroup which is our main object of interest in the present paper. We prove that for each set of boundary parameters, the dynamics generator has a compact inverse and this inverse operator belongs to class \documentclass{article}\pagestyle{empty}\usepackage{amsfonts}\begin{document}$\mathfrak{S}_p$\end{document} of compact operators with p > 1. We also show that if both boundary parameters are not purely imaginary numbers, then the dynamics generator is a nonselfadjoint operator in the energy space. However, its inverse operator is a finite-rank perturbation of a selfadjoint operator. The latter fact is crucial for the proof of the fact that the root vectors of the dynamics generator form a complete and minimal set in the energy space. We will use the spectral results in our forthcoming papers to prove that the dynamics generator of the system is a Riesz spectral operator in the sense of Dunford and to use the latter fact for the solution of several boundary and distributed controllability problems via the spectral decomposition method. [ABSTRACT FROM AUTHOR]