Stratification and bundle structure of the set of conditioned invariant subspaces in the general case

Abstract

We extend some known results on the smooth stratification of the set of conditioned invariant subspaces to a general pair (C,A)∈ C (p+n)×n without any assumption on the observability. More precisely, we prove that the set of ( C, A)-conditioned invariant subspaces having a fixed Brunovsky–Kronecker structure is a submanifold of the corresponding Grassmann manifold, with a fiber bundle structure relating the observable and nonobservable part, and we then compute its dimension. We also prove that the set of all ( C, A)-conditioned invariant subspaces having a fixed dimension is connected, provided that the nonobservable part of ( C, A) has at most one eigenvalue (this condition is in general necessary).