Differentiable structure of the set of controllable (A,B)t-invariant subspaces

On the Geometry of the Solutions of the Cover Problem

“…But 1 = 2 = 3 = (1); and so either 3 ∩ n 1 = ∅ or 2 ∩ n 2 = ∅. That is to say there is not (A, B)-controlled invariant subspace V such that the restriction of (B t (2), (3), (4,5)). Since 3−j+1 ∩ n j = ∅ for j = 1, 2, 3 we conclude that there is at least one (A, B)-invariant subspace of codimension 7 containing S. The orthogonal of such a subspace is generated by a matrix Y ∈ M(r, s) in reduced form as given in Definition 2.5.…”

“…In order to prove (a) notice that if Γ 1 = {(X, X ) ∈ M × M|X = XP, P ∈ G} then Γ = Γ 1 ∩ (N × N ). As Γ 1 is closed in M × M (see [5]), Γ is closed in N × N .…”

“…, k. In other words h and k are compatible if and only if s and r are compatible. In the sequel we also assume that s and r are compatible partitions and we denote by Inv (r, s) the set of (C, A)-conditioned invariant subspaces for which the restriction of (C, A) to each one of them has Brunovsky indices s (see [5] or [6] for the definition of this restriction).…”

Conditioned Invariant Subspaces and the Geometry of Nilpotent Matrices

“…of (C, A)-invariant subspaces with prescribed dimension k allows a stratification into smooth manifolds, the so-called Brunovsky-Kronecker strata [6]. However, it is still unclear whether Inv k (C, A) is a smooth manifold itself.…”

“…spectral factorization, linear quadratic control, H ∞ and game theory, as well as observer theory, filtering and estimation. However, it is only until recently, that first attempts have been made towards a better understanding of the geometry of the set of conditioned invariant subspaces Inv k (C, A); see [9], [10], [19], [6], [17]. The recent Ph.D. thesis [29] contains a comprehensive summary.…”