Optimization and Pseudospectra, with Applications to Robust Stability
James V. Burke, Adrian S. Lewis, Michael L. Overton · SIAM Journal on Matrix Analysis and Applications · 2003
The $\epsilon$-pseudospectrum of a matrix A is the subset of the complex plane consisting of all eigenvalues of all complex matrices within a distance $\epsilon$ of A. We are interested in two aspects of "optimization and pseudospectra." The first concerns maximizing the function "real part" over an $\epsilon$-pseudospectrum of a fixed matrix: this defines a function known as the $\epsilon$-pseudospectral abscissa of a matrix. We present a bisection algorithm to compute this function. Our second interest is in minimizing the $\epsilon$-pseudospectral abscissa over a set of feasible matrices. A prerequisite for local optimization of this function is an understanding of its variational properties, the study of which is the main focus of the paper. We show that, in a neighborhood of any nonderogatory matrix, the $\epsilon$-pseudospectral abscissa is a nonsmooth but locally Lipschitz and subdifferentially regular function for sufficiently small $\epsilon$; in fact, it can be expressed locally as the maximum of a finite number of smooth functions. Along the way we obtain an eigenvalue perturbation result: near a nonderogatory matrix, the eigenvalues satisfy a Hölder continuity property on matrix space---a property that is well known when only a single perturbation parameter is considered. The pseudospectral abscissa is a powerful modeling tool: not only is it a robust measure of stability, but it also reveals the transient (as opposed to asymptotic) behavior of associated dynamical systems.