We develop spectral and asymptotic analysis for a class of nonselfadjoint operators which are the dynamics generators for the systems governed by the equations of the spatially nonhomogeneous Timoshenko beam model with a 2–parameter family of dissipative boundary conditions. Our results split into two groups. We prove asymptotic formulas for the spectra of the aforementioned operators (the spectrum of each operator consists of two branches of discrete complex eigenvalues and each branch has only two points of accumulation: +∞ and —∞), and for their generalized eigenvectors. Our second main result is the fact that these operators are Riesz spectral. To obtain this result, we prove that the systems of generalized eigenvectors form Riesz bases in the corresponding energy spaces. We also obtain the asymptotics of the spectra and the eigenfunctions for the nonselfadjoint polynomial operator pencils associated with these operators. The pencil asymptotics are essential for the proofs of the spectral results for the aforementioned dynamics generators.
We consider a class of nonselfadjoint quadratic operator pencils generated by the equation, which governs the vibrations of a string with nonconstant bounded density subject to viscous damping with a nonconstant damping coefficient. These pencils depend on a complex parameter h, which enters the boundary conditions. Depending on the values of h, the eigenvalues of the above pencils may describe the resonances in the scattering of elastic waves on an infinite string or the eigenmodes of a finite string. We obtain the 7asymptotic representations for these eigenvalues. Assuming that the proper multiplicity of each eigenvalue is equal to one, we prove that the eigenfunctions of these pencils form Riesz bases in the weighted L2-space, whose weight function is exactly the density of the string. The general case of multiple eigenvalues will be treated in another paper, based on the results of the present work. 1.Introduction.In the present paper we develop the spectral analysis for a one-parameter family of nonselfadjoint quadratic operator pencils. These pencils appear in the study of the wave equation which describes a nonhomogeneous damped string. The main result of the work is the proof of the fact that the systems of the eigenvectors and associated vectors of the above mentioned pencils form unconditional bases in the corresponding Hilbert space. There exists an extensive literature on operator pencils (it can be traced, for example, through the references in the book [1]). However, known abstract results cannot be applied to the pencils we consider in this work.In the present paper we prove the above result in the case when there are no associated vectors, i.e. the proper multiplicity (see [1,2]) of each eigenvalue is equal to one. The proof of the main result in the general case will be given in another paper. This proof will be based on the results of the present work. We have considered the case of
The present paper is devoted to the asymptotic and spectral analysis of a model of an aircraft wing in a subsonic air flow. This model has been developed in the Flight Systems Research Center of the University of California at Los Angeles. 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 self-straining actuators. The differential parts of these equations form a coupled linear hyperbolic system; the integral parts are of the convolution type. The system of equations of motion is equivalent to a single operator evolution-convolution equation in the energy space. The Laplace transform of the solution of this equation can be represented in terms of the so-called generalized resolvent operator, which is an operator-valued function of the spectral parameter. This generalized resolvent operator is a finite meromorphic function on the complex plane having a branch cut along the negative real semi-axis. Its poles are precisely the aeroelastic modes, which are the main object of interest in the present paper. The dynamics generator of the differential part of the system has been systematically studied in a series of works by the second author. This generator is a non-self-adjoint operator in the energy space with a purely discrete spectrum. In the aforementioned series of papers, it was shown that the spectrum consists of two branches, and the precise spectral asymptotics with respect to the eigenvalue number was derived. The asymptotic approximations for the mode shapes have also been obtained. Based on the asymptotic results, it has been proved that the set of the generalized eigenvectors of the dynamics generator forms a Riesz basis in the energy space. In the present paper, we consider the entire integrodifferential system which governs the model. Namely, we investigate the properties of the integral convolution-type part of the original system. We show, in particular, that the set of the aeroelastic modes is asymptotically close to the discrete spectrum of the dynamics generator corresponding to the differential part. The results of this paper will be important for the reconstruction of the solution of the original initialboundary-value problem from its Laplace transform and for the analysis of the flutter phenomenon in the forthcoming work.
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