Riesz basis property of root vectors of nonselfadjoint operators generated by radial damped wave equations

Abstract

We consider an infinite sequence of radial wave equations obtained by the separation of variables in the spherical coordinates from the 3-dimensional damped wave equation with spatially nonhomogeneous spherically symmetric coefficients. Our main objects of interest are the nonselfadjoint operators in the energy spaces of 2-component initial data, which are the dynamics generators for the systems governed by the aforementioned equations and nonselfadjoint boundary conditions on the sphere $r=a$. Our main result is the fact that the sets of the root vectors (generalized eigenvectors) of these operators form Riesz bases in the corresponding energy spaces. This result has several applications. The first one is the fact that the aforementioned operators are spectral in the sense of N. Dunford, and, therefore, we have a new nontrivial class of spectral operators. Another application is a precise estimate on the rate of the energy decay, which is equal to the spectral abscissa of the corresponding semigroup. Finally, we use the results of our spectral analysis to formulate the solutions of several problems in control theory of systems governed by damped wave equations (the proofs are given in another work).