Measuring and Optimizing Stability of Matrices

Michael Overton
Courant Institute of Mathematical Sciences
New York University
New York, NY, USA

A matrix is stable if its spectral abscissa (maximum real part of its eigenvalues) is negative. The spectral abscissa models only asymptotic behavior of associated dynamical systems, so more practical stability measures include the pseudospectral abscissa (maximum real part of the pseudospectrum) and the distance to instability (minimum norm perturbation required to make a stable matrix unstable). Matrices often arise in applications as parameter dependent, a classic example being A(K) = A0 + BKC, the static output feedback model in control. We formulate two optimization problems over a parameterized matrix family: minimization of the pseudospectral abscissa and maximization of the distance to instability, and present a new algorithm that approximates local optimizers. One of our numerical examples is a difficult stabilization problem from the control literature: a model of a Boeing 767 at a flutter condition.

Joint work with James V. Burke, University of Washington, Seattle, WA and Adrian S. Lewis, Simon Fraser University, Burnaby, BC, Canada.

Brief Biography: Michael L. Overton received his BSc from UBC in 1974, along with the Governor General's Gold Medal for Arts and Sciences. He received the MS and PhD degrees in Computer Science from Stanford University. He is currently Professor of Computer Science and Mathematics at the Courant Institute of Mathematical Sciences, New York University. Michael Overton is an elected member of the Board of Trustees of SIAM (Society for Industrial and Applied Mathematics) and has also served on the SIAM Council. He is a member of the Council of FoCM (Foundations of Computational Mathematics) and of the Board of Directors of the Canadian Mathematical Society. He serves on the editorial boards of SIAM Journal on Optimization (for which he was Editor-in-Chief from 1995-1999), SIAM Journal on Matrix Analysis and Applications, the IMA Journal on Numerical Analysis, and SIAM Review. His research interests are at the interface of optimization and linear algebra, especially nonsmooth optimization problems involving eigenvalues, with applications to many different subjects including robust control, structural analysis, combinatorial optimization and convex analysis. He is author of "Numerical Computing with IEEE Floating Point Arithmetic" (SIAM, 2001).