Robust stability and a criss-cross algorithm for pseudospectra

A dynamical system (x) over dot = Ax is robustly stable when all eigenvalues of complex matrices within a given distance of the square matrix A lie in the left half-plane. The 'pseudospectral abscissa', which is the largest real part of such an eigenvalue, measures the robust stability of A. We present an algorithm for computing the pseudospectral abscissa, prove global and local quadratic convergence, and discuss numerical implementation. As with analogous methods for calculating H-infinity norms, our algorithm depends on computing the eigenvalues of associated Hamiltonian matrices.