@article{3fc98360a51941f88de9c613f9328b6c,
title = "Robust stability and a criss-cross algorithm for pseudospectra",
abstract = "A dynamical system ẋ = 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∞ norms, our algorithm depends on computing the eigenvalues of associated Hamiltonian matrices.",
keywords = "Eigenvalue optimization, H norm, Hamiltonian matrix, Pseudospectrum, Robust control, Robust optimization, Spectral abscissa, Stability",
author = "Burke, {J. V.} and Lewis, {A. S.} and Overton, {M. L.}",
note = "Funding Information: Many thanks to Carsten Scherer for pointing out Boyd & Balakrishnan (1990), setting us on the course that led to this work. Thanks also to Peter Benner for providing the Hamiltonian eigenvalue software and to Emre Mengi for providing a MATLAB interface for the complex Hamiltonian code. We particularly thank Nick Trefethen and Tom Wright for their enthusiasm for adding the criss-cross algorithm to EigTool, and both Emre Mengi and Tom Wright for carrying this out in short order. The first author{\textquoteright}s research was supported in part by National Science Foundation Grant DMS-0203175. The second author{\textquoteright}s research was supported in part by NSERC. The third author{\textquoteright}s research was supported in part by National Science Foundation Grant CCR-0098145.",
year = "2003",
month = jul,
doi = "10.1093/imanum/23.3.359",
language = "English (US)",
volume = "23",
pages = "359--375",
journal = "IMA Journal of Numerical Analysis",
issn = "0272-4979",
publisher = "Oxford University Press",
number = "3",
}