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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 present paper is devoted to mathematical analysis of the model that describes fluid flow moving in a channel with flexible walls, which are subject to traveling waves. Experimental data show that the energy of the flowing fluid can be consumed by the structure (the walls) inducing "traveling wave flutter." In the problems involving two‐media interactions (fluid/structure), flutter‐like perturbations can occur either in the fluid flowing in the channel with harmonically moving walls, or in the solid structure interacting with the flow. In the present research, it is shown that there are no abrupt (or flutter‐like) changes in the flow velocity profiles. Using the mass conservation law and incompressibility condition, we obtain the initial boundary value problem for the stream function. The boundary conditions reflect that (i) there is no movement in the vertical direction along the axis of symmetry and (ii) there is no relative movement between the near‐boundary flow and the structure ("no‐slip" condition). The closed form solution is derived for the stream function, which is represented in the form of an infinite functional series. [ABSTRACT FROM AUTHOR]

Three control problems for the system of two coupled differential equations governing the dynamics of an energy harvesting model are studied. The system consists of the equation of an Euler–Bernoulli beam model and the equation representing the Kirchhoff's electric circuit law. Both equations contain coupling terms representing the inverse and direct piezoelectric effects. The system is reformulated as a single evolution equation in the state space of 3‐component functions. The control is introduced as a separable forcing term g(x)f(t) on the right‐hand side of the operator equation. The first control problem deals with an explicit construction of f(t) that steers an initial state to zero on a time interval [0, T]. The second control problem deals with the construction of f(t) such that the voltage output is equal to some given function v(t) (with g(x) being given as well). The third control problem deals with an explicit construction of both the force profile, g(x), and the control, f(t), which generate the desired voltage output v(t). Interpolation theory in the Hardy space of analytic functions is used in the solution of the second and third problems. [ABSTRACT FROM AUTHOR]

The Euler‐Bernoulli beam model with fully nonconservative boundary conditions of feedback control type is investigated. The output vector (the shear and the moment at the right end) is connected to the observation vector (the velocity and its spatial derivative on the right end) by a 2 × 2 matrix (the boundary control matrix), all entries of which are nonzero real numbers. For any combination of the boundary parameters, the dynamics generator, L, of the model is a non–self‐adjoint matrix differential operator in the state Hilbert space. A set of 4 self‐adjoint operators, defined by the same differential expression as L on different domains, is introduced. It is proven that each of these operators, as well as L, is a finite‐rank perturbation of the same self‐adjoint dynamics generator of a cantilever beam model. It is also shown that the non–self‐adjoint operator, L, shares a number of spectral properties specific to its self‐adjoint counterparts, such as (1) boundary inequalities for the eigenfunctions, (2) the geometric multiplicities of the eigenvalues, and (3) the existence of real eigenvalues. These results are important for our next paper on the spectral asymptotics and stability for the multiparameter beam model. [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]

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]