Smallest destabilizing perturbations for linear systems

Abstract Necessary and sufficient conditions on a perturbation £ that will guarantee the asymptotic stability of the linear system [xdot] = (A + E)x given that the nominal system [xdot] = Ax is asymptotically stable are derived. The class of all perturbations E of minimal norm that will destabilize the nominal system is characterized in terms of the norm of the resolvent matrix (iωI — A)−1 for an appropriate ω ∊ R These results are specialized to the euclidean norm and expressed in terms of singular values. Analogous results are also obtained for the difference equation x(k + 1) = Ax(k) (k = 0,1,2,.)