Spectral Operators Generated by 3-Dimensional Damped Wave Equation and Applications to Control Theory

Abstract

We formulate our results on the spectral analysis for a class of nonselfadjoint operators in a Hilbert space and on the applications of this analysis to the control theory of linear distributed parameter systems. The operators, we consider, are the dynamics generators for systems governed by 3-dimensional wave equation which has spacially nonhomogeneous spherically symmetric coefficients and contains a first order damping term. We consider this equation with a one-parameter family of linear first order boundary conditions on a sphere. These conditions contain a damping term as well. Our main object of interest is the class of operators in the energy space of 2-component initial data which generate the dynamics of the above systems. Our first main result is the fact that these operators are spectral in the sense of N. Dunford. This result is obtained as a corollary of two groups of results: (i) asymptotic representations for the complex eigenvalues and eigenfunctions, and (ii) the fact that the systems of eigenvectors and associated vectors form Riesz bases in the energy space. We also present an explicit solution of the controllability problem for the distributed parameter systems governed by the aforementioned equation using the spectral decomposition method.