Marianna A. Shubov

Asymptotic and Spectral Analysis of the Spatially Nonhomogeneous Timoshenko Beam Model

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.

Basis property of eigenfunctions of nonselfadjoint operator pencils generated by the equation of nonhomogeneous damped string

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 parameterh, which enters the boundary conditions. Depending on the values ofh, the eigenvalues of the above pencils may describe the resonances in the scattering of elastic waves on an infinite string or the eingenmodes 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 weightedL2-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.

Spectral operators generated by damped hyperbolic equations

We announce a series of results on the spectral analysis for a class of nonselfadjoint opeators, which are the dynamics generators for the systems governed by hyperbolic equations containing dissipative terms. Two such equations are considered: the equation of nonhomogeneous damped string and the 3-dimensional damped wave equation with spacially nonhomogeneous spherically symmetric coefficients. Nonselfadjoint boundary conditions are imposed at the ends of a finite interval or on a sphere centered at the origin respectively. Our main result is the fact the aforementioned operators are spectral in the sense of N. Dunford. The result follows from the fact that the systems of root vectors of the above operators form Riesz bases in the corresponding energy spaces. We also give asymptotics of the spectra and state the Riesz basis property results for the nonselfadjoint operator pencils associated with these operators.

Spectral operators generated by Timoshenko beam model

We announce a series of results on the spectral analysis for a class of nonselfadjoint operators which are the dynamics generators for the systems governed by the equations of the Timoshenko beam model with a 2-parameter family of dissipative boundary conditions. Our results split into three groups. (1) We present asymptotic formulas for the spectra of the aforementioned operators (the spectrum of each operator consists of two branches of discrete complex eigenvalues) and for their generalized eigenvectors. (2) We show that these operators are Riesz spectral. This result follows from the fact that the systems of generalized eigenvectors form Riesz bases in the corresponding energy spaces. (3) We give the asymptotics of the spectra and the eigenfunctions for the nonselfadjoint polynomial operator pencils associated with these operators. Our results, on one hand, provide a class of nontrivial examples of spectral operators (nonselfadjoint operators which admit an analog of spectral decomposition). On the other hand, these results give a key to the solutions of various control and stabilization problems for the Timoshenko beam model using the spectral decomposition method.

Exact Controllability of the Damped Wave Equation

We study the controllability problem for a distributed parameter system governed by the damped wave equation

Riesz basis property of root vectors of non-self-adjoint operators generated by aircraft wing model in subsonic airflow

u_{tt}-\frac{1}{\rho(x)}\frac{d}{dx}\left(p(x)\frac{du}{dx}\right)+ 2d(x)u_t+q(x)u=g(x)f(t),

On controllability of an elastic string with a viscous damping

where

Asymptotics of aeroelastic modes and basis property of mode shapes for aircraft wing model

x\in (0,a)