How near is a stable matrix to an unstable matrix?
Charles F. Van Loan · Contemporary mathematics - American Mathematical Society · 1985
In this paper we explore how close a given stable matrix A is to being unstable. As a measure of "how stable" a stable matrix is, the spectral abscissa is shown to be flawed. A better measure of stability is the Frobenius norm of the smallest perturbation that shifts one of A's eigenvalues to the imaginary axis. This leads to a singular value minimization problem that can be approximately solved by heuristic means. However, the minimum destabilizing perturbation may be complex even when A is real. This suggests that in the real case we look for the smallest real perturbation that shifts one of the eigenvalues to the imaginary axis. Unfortunately, a difficult constrained minimization problem ensues and no practical estimation technique could be devised.