{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,18]],"date-time":"2026-09-18T05:44:16Z","timestamp":1789710256037,"version":"4.0.1"},"reference-count":44,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2003,1]]},"abstract":"<jats:p>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 H\u00f6lder 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.<\/jats:p>","DOI":"10.1137\/s0895479802402818","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"80-104","source":"Crossref","is-referenced-by-count":48,"title":["Optimization and Pseudospectra, with Applications to Robust Stability"],"prefix":"10.1137","volume":"25","author":[{"given":"J. V.","family":"Burke","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"A. S.","family":"Lewis.","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"M. 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Overton,\n                      Robust stability and a criss\u2010cross algorithm for pseudospectra\n                      , IMA J. Numer. Anal., to appear."},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(02)00260-4"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1007\/s102080010008"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1137\/0909059"},{"key":"R17","volume-title":"Optimization and nonsmooth analysis","author":"Clarke Frank","year":"1983"},{"key":"R18","unstructured":"Y. Genin, P. Van Dooren, and V. Vermaut,\n                      Convergence of the calculation of H\u221e norms and related questions\n                      , in Proceedings MTNS\u201098, G. Picci and D. S. 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