Asymptotic analysis of stabilizability of a control system for a discretized boundary damped wave equation

This paper attempts to answer an open question on the uniform stabilizability of a commonly used finite difference method for the approximation of a control system modeled by 1-dimensional weakly damped wave equation. A detailed analysis of the spectral properties of the matrices used in the approximation and the asymptotic properties of the eigenvalues as a function of the dimension of the approximation space allow to conclude that finite dimensional control systems given by the approximation method considered here are not uniformly stabilizable. The results on the spectral properties of the matrices used in the finite difference approximation of the partial differential equations discussed in this paper also provide a method to calculate each individual eigenvalues of the matrix through a scalar iterative method. It is used in this paper to obtain accurate estimates on the stability and stabilizability margins of the finite dimensional control systems as well as to compute the eigenvalues numerically to verify our theoretical results.