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The distribution of natural frequencies of the Euler–Bernoulli beam subject to fully non-dissipative boundary conditions is investigated. The beam is clamped at the left end and equipped with a 4-parameter (⁠|$\alpha ,\beta ,k_1,k_2$|⁠) linear boundary feedback law at the right end. The |$2 \times 2$| boundary feedback matrix relates the control input (a vector of velocity and its spatial derivative at the right end), to the output (a vector of shear and moment at the right end). The initial boundary value problem describing the dynamics of the beam has been reduced to the first order in time evolution equation in the state Hilbert space equipped with the energy norm. The dynamics generator has a purely discrete spectrum (the vibrational modes) denoted by |$\{\nu _n\}_{n\in \mathbb {Z}^{\prime}}$|⁠. The role of the control parameters is examined and the following results have been proven: (i) when |$\beta \neq 0$|⁠ , the set of vibrational modes is asymptotically close to the vertical line on the complex |$\nu$| -plane given by the equation |$\Re \nu = \alpha + (1-k_1k_2)/\beta$|⁠ ; (ii) when |$\beta = 0$| and the parameter |$K = (1-k_1 k_2)/(k_1+k_2)$| is such that |$\left |K\right |\neq 1$| then the following relations are valid: |$\Re (\nu _n/n) = O\left (1\right)$| and |$\Im (\nu _n/n^2) = O\left (1\right)$| as |$\left |n\right |\to \infty$|⁠ ; (iii) when |$\beta =0$|⁠ , |$|K| = 1$|⁠ , and |$\alpha = 0$|⁠ , then the following relations are valid: |$\Re (\nu _n/n^2) = O\left (1\right)$| and |$\Im (\nu _n/n) = O\left (1\right)$| as |$\left |n\right |\to \infty$|⁠ ; (iv) when |$\beta =0$|⁠ , |$|K| = 1$|⁠ , and |$\alpha>0$|⁠ , then the following relations are valid: |$\Re (\nu _n/\ln \left |n\right |) = O\left (1\right)$| and |$\Im (\nu _n/n^2) = O\left (1\right)$| as |$\left |n\right |\to \infty$|⁠. [ABSTRACT FROM AUTHOR]

The solution of the pole reassignment control problem for a distributed parameter system is presented. The system is a vibrating string with spatially non-homogeneous density and modulus of elasticity coefficients and with both distributed and boundary damping. The string equation and the boundary conditions are reduced to a single linear evolution equation in the Hilbert space of two-component Cauchy data equipped with energy metric. It is the dynamics generator of this equation, whose spectrum is rearranged with a feedback control. The solution is based on the fact that the dynamics generator, its finite-rank perturbations and the adjoints to these operators are Riesz spectral, i.e. their sets of the generalized eigenvectors form (non-orthogonal) Riesz bases in the state space. The approach extends to any system, whose dynamics generator has similar properties. [ABSTRACT FROM PUBLISHER]

The zero controllability problem for the hyperbolic equation, which governs the vibrations of the Euler–Bernoulli beam model of a finite length, is studied in this paper. The equation of motion is supplied with a one-parameter family of physically meaningful boundary conditions containing damping terms. The control is introduced as a separable forcing term g(x)f(t) in the right-hand side of the equation. A force profile function, g(x), is assumed to be given. To construct the control, f(t), which brings a given initial state of the system to zero on the specific time interval [0, T], the spectral decomposition method is applied. The necessary and/or sufficient conditions for the exact controllability as well as the explicit formulas for the control laws are given. Approximate controllability is also discussed. The exact controllability results are generalized to the case of multiple eigenvalues of the main dynamics generator. [ABSTRACT FROM PUBLISHER]

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]

We study the zero controllability problem for the system of two coupled hyperbolic equations which govern the vibrations of the Timoshenko beam with spatially nonhomogeneous coefficients: mass density, flexural rigidity and shear stiffness. The system is considered on a finite interval with a two-parameter family of boundary conditions containing damping terms. The controls are introduced as separable forcing terms gi(x)fi(t), i = 1,2, in the right-hand sides of both equations. We apply the spectral decomposition method 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 force profile functions, gi(x), i = 1,2, are assumed to be given. Our approach is based on the results obtained in the recent work by the author. In this work, a detailed asymptotic and spectral analysis of the nonselfadjoint operator, which generates the dynamics of the Timoshenko beam with the dissipative boundary conditions, was given. It was shown that the aforementioned operator was Riesz spectral, i.e. its generalized eigenvectors formed a Riesz basis in the energy space. Explicit asymptotic formulas for two branches of the Timoshenko beam spectrum were also presented. Based on these spectral results, we reduce the control problem to the corresponding moment problem. We solve this moment problem using the asymptotics of the spectrum and give explicit formulas for the controls. [ABSTRACT FROM PUBLISHER]

This paper is the first in a series of several works devoted to the asymptotic and spectral analysis of an aircraft wing in a subsonic air flow. This model has been developed in the Flight Systems Research Center of UCLA and is presented in the works of Balakrishnan. The model is governed by a system of two coupled integro-differential equations and a two parameter family of boundary conditions modelling the action of the self-straining actuators. The unknown functions (the bending and the torsion angle) depend on time and one spatial variable. The differential parts of the above equations form a coupled linear hyperbolic system; the integral parts are of convolution type. The system of equations of motion is equivalent to a single operator evolution–convolution type equation in the state space of the system equipped with the so-called energy metric. The Laplace transform of the solution of this equation can be represented in terms of the so-called generalized resolvent operator. The generalized resolvent operator is an operator-valued function of the spectral parameter. This generalized resolvent operator is a finite meromorphic function defined on the complex plane having the branch cut along the negative real semi-axis. The poles of the generalized resolvent are precisely the aeroelastic modes, and the residues at these poles are the projectors on the generalized eigenspaces. In this paper, our main object of interest is the dynamics generator of the differential parts of the system. It is a non-selfadjoint operator in the state space with a pure discrete spectrum. In the present paper, we show that the spectrum consists of two branches, and we derive their precise spectral asymptotics. Based on these results, in the next paper we will derive the asymptotics of the aeroelastic modes and approximations for the mode shapes. [ABSTRACT FROM AUTHOR]