Optimization of Pseudospectra

New York University, New York NY

Investigators

Abstract

Optimization problems involving eigenvalues arise in many applications. In recent years, attention has focused on semidefinite programs, which are linear optimization problems in the space of real symmetric matrices, with positive semidefinite constraints. This project focuses on optimization problems in the larger space of square matrices, not generally symmetric. In this context, it is well known that eigenvalues, which provide information about asymptotic behavior of dynamical systems, are not as useful as pseudospectra, which provide a more robust measure of system behavior. This project focuses on the computation, analysis and, especially, numerical optimization of pseudospectral functions. A secondary goal is to advance the development of algorithms for general nonsmooth, nonconvex optimization problems. Applications of this work arise in many contexts. A good example that we have used is a model of a Boeing 767 at a flutter condition. Flutter occurs when the plane flies so fast that the interaction of the aerodynamic (wind) forces with the structural forces in the airplane combine to generate an instability which could be catastrophic. Normally, a plane like this does not fly so fast but it is important to know what would happen if it did, and whether a simple controller could bring the plane under control. Eigenvalues tell us about stability in the long term, but pseudospectra are far more informative, telling us about stability in the short term. Optimizing the pseudospectra over the parameters of the controller may result in the design of a controller that is simple yet effective in avoiding the instability and preventing catastrophe.