The Riesz basis property of the system of root vectors for the equation of a nonhomogeneous damped string: Transformation operator method. (English) Zbl 0986.74039

Shubov, Marianna A.

Methods Appl. Anal. 6, No. 4, 571-591 (1999).

From the summary: We prove the spectral decomposition theorem for a class of nonselfadjoint operators in Hilbert space. These operators are the dynamics generators for the systems governed by one-dimensional hyperbolic equations with spatially nonhomogeneous coefficients containing first-order damping terms and subject to linear nonselfadjoint boundary conditions. These equations and boundary conditions describe, in particular, a spatially nonhomogeneous string subject to a distributed viscous damping and damped at the boundary points. Our main result leading to the spectral decomposition is the fact that the root vectors (eigenvectors and associated vectors together) of the above operators form Riesz bases in the corresponding energy spaces.

MSC:

74H45	Vibrations in dynamical problems in solid mechanics
74K05	Strings
35P10	Completeness of eigenfunctions and eigenfunction expansions in context of PDEs

Keywords:

eigenvectors; spectral decomposition theorem; nonselfadjoint operators; Hilbert space; one-dimensional hyperbolic equations; first-order damping; linear nonselfadjoint boundary conditions; nonhomogeneous string; distributed viscous damping; root vectors; Riesz basis; energy spaces