Fast algorithms for computing the distance to instability of nonlinear eigenvalue problems, with application to time-delay systems.

A continuous dynamical system is stable if all eigenvalues lie strictly in the left half of the complex plane. However, this is not a robust measure because stability is no longer guaranteed when the system parameters are slightly perturbed. Therefore, the pseudospectrum of a matrix and its pseudospectral abscissa are studied. Mostly, one is often interested in computing the distance to instability, because it is a robust measure for stability against perturbations. % As a first contribution, this paper presents two iterative methods for computing the distance to instability, considering complex perturbations. The first one is based on locating a zero of the pseudospectral abscissa function. This method is particularly suitable for large sparse matrices as it is based on repeated eigenvalue computations, where the original matrix is perturbed with a rank one matrix. The second method is based on a recently proposed global optimization technique. The advantages of both methods can be combined in a hybrid algorithm. As a second contribution we show that the methods apply to a broad class of nonlinear eigenvalue problems, in particular eigenvalue problems inferred from linear delay-differential equations, and, therefore, they are useful for a wide range of problems. In the numerical examples the standard eigenvalue problem, the quadratic eigenvalue problem and the delay eigenvalue problem are addressed.