ESAIM: Control, Optimisation and Calculus of VariationsWill

We consider a problem in eigenvalue optimization, in particular finding a local minimizer of the spectral abscissa the value of a parameter that results in the smallest magnitude of the largest real part of the spectrum of a matrix system. This is an important problem for the stabilization of control systems. Many systems require the spectra to lie in the left half plane in order for stability to hold. The optimization problem, however, is difficult to solve because the underlying objective function is nonconvex, nonsmooth, and non-Lipschitz. In addition, local minima tend to correspond to points of non-differentiability and locally non-Lipschitz behavior. We present a sequential linear and quadratic programming algorithm that solves a series of linear or quadratic subproblems formed by linearizing the surfaces corresponding to the largest eigenvalues. We present numerical results comparing the algorithms to the state of the art. 1991 Mathematics Subject Classification. 90C55, 90C90, 93D21, 90C26, 35P30, 47J10.