Accurate solutions of ill-posed problems in control theory

We present computable, guaranteed error bounds for controllable subspaces and uncontrollable modes, unobservable subspaces and unobservable modes, supremal (A,C) invariant subspaces in ker D, supremal (A,C) controllable subspaces in ker D, the uncontrollable modes within the supremal (A,C) invariant subspace in ker D, and invariant zeroes. In particular our bounds apply in the nongeneric case when the solutions are ill-posed. We do this by showing that all these features are eigenspaces and eigenvalues of certain singular matrix pencils, and then applying a perturbation theory for general singular matrix pencils. Numerical examples are included.