On the Lipschitz stability of (A,B)-invariant subspaces

Let T be the set of triples (A,B,S) where S is an (A, B)-invariant subspace and let Fμ be the matrix having minimum norm such that (A+BFμ)S)⊂S. Then, if θ:T→Mm,n is the map defined by θ(A,B,S)=Fμ and θ is continuous at (A,B,S) a simple necessary and sufficient condition is given for the Lipschitz stability of S. It is shown that this continuity condition is satisfied in an open and dense subset of T and that the set of triples (A,B,S) such that S is Lipschitz stable is also open and dense in T.