EECS Joint Colloquium Distinguished Lecture Series

Wednesday, October 22, 2003
Hewlett Packard Auditorium, 306 Soda Hall
4:00-5:00 p.m.

Dr. Michael Overton

Courant Institute of Mathematical Sciences,
New York University


Optimizing Matrix Stability and Controllability




A matrix is stable if its eigenvalues are in the left half of the complex plane. 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). Likewise, the classical definition of controllability is not as useful as a measure of the distance to uncontrollability.

Matrices often arise in applications as parameter dependent. Optimization of stability or controllability measures over parameters is challenging because the objective functions are nonsmooth and nonconvex. We solve such optimization problems, locally at least, via a novel method based on gradient sampling. One of our stability optimization examples is a difficult problem from the control literature: finding stable low-order controllers for a model of a Boeing 767 at a flutter condition. We also give a controllability optimization example and explain its connection with an interesting open question: how many connected components are possible for pseudospectra of rectangular matrices?


Michael L. Overton is a Professor of Computer Science and Mathematics at the Courant Institute of Mathematical Sciences, New York University, where he has worked since receiving his Ph.D. from Stanford in 1979. He is editor-in-chief of the MPS-SIAM book series and was editor-in-chief of SIAM Journal on Optimization from 1995 to 1999. He also serves on the editorial boards of various other journals, and is an an elected member of the SIAM Board of Trustees. His research interests are at the interface of optimization and linear algebra, especially nonsmooth optimization problems involving eigenvalues. He is also the author of "Numerical Computing with IEEE Floating Point Arithmetic", published by SIAM in 2001.