Optimization and Pseudospectra, with Applications to Robust Stability

The -pseudospectrum of a matrix A is the subset of the complex plane consisting of all eigenvalues of all complex matrices within a distance of A. We are interested in two aspects of “optimization and pseudospectra.” The first concerns maximizing the function “real part” over an -pseudospectrum of a fixed matrix: this defines a function known as the -pseudospectral abscissa of a matrix. We present a bisection algorithm to compute this function. Our second interest is in minimizing the -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 -pseudospectral abscissa is a nonsmooth but locally Lipschitz and subdifferentially regular function for sufficiently small ; 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.