Nonsmooth optimization and robust control

Many questions of robust control analysis and synthesis fundamentally involve nonsmooth sets and functions, and their variational properties. Central examples include distances to instability and uncontrollability, the H∞H∞ norm, and pseudospectra. This plenary presentation at the Fifth IFAC Symposium on Robust Control Design (2006), surveys what current ideas from nonsmooth analysis say about the structure and conditioning of such functions and sets, and their numerical optimization. The presentation focuses on notions of nonsmooth derivatives and regularity and on structural tools such as partial smoothness and semi-algebraic techniques, illustrating how each idea helps in analysis and algorithm design. This survey relies heavily on joint work of the author with J.V. Burke (University of Washington) and M.L. Overton (Courant Institute).