Xavier Puerta

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

Abstract Given (A,B) t ∈ Hom( C n , C n+m ) observable, we prove that the set of ( A,B ) t -invariant subspaces having a fixed Brunovsky-Kronecker structure is a connected manifold, and we compute its dimension. Also, we include some applications of these results.

The topology of the set of conditioned invariant subspaces

We consider the topology of the set of conditioned invariant subspaces of an observable pair

GEOMETRIC CHARACTERIZATION AND CLASSIFICATION OF MARKED SUBSPACES

(C,A)

On the geometry of the set of controllability subspaces of a pair (A,B)

of a fixed dimension. By fixing the observability indices of the restricted system, a stratification by finitely many smooth manifolds is obtained, termed Brunovsky strata. It is shown that each Brunovsky stratum is homotopy equivalent to a generalized flag manifold. From this description an effective formula for the Betti numbers of the Brunovsky strata can be derived.

Global reduction to the Kronecker canonical form of a Cr-family of time-invariant linear systems☆

Abstract Given an endomorphism of a finite dimensional C -vector space, we obtain geometric conditions in order to characterize and to classify the marked invariant subspaces W ⊂ E (that is to say, the invariant subspaces having some Jordan basis which can be extended to a Jordan basis of E ). Both conditions are expressed in terms of the double family of subspaces (Ker f h ∩ Im f d ). The starting point is a new criterion for extensibility of Jordan bases.

A coordinate atlas of the manifold of observable conditioned invariant subspaces

Abstract Given a controllable system defined by a pair of matrices (A,B), we investigate the geometry of the set of controllability subspaces. This set is a subset of the set of (A,B)-invariant subspaces. We prove that, in fact, it is a stratified submanifold and we compute its dimension.

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

Abstract We present an approach to the Kronecker equivalence of quadruples of matrices (A, B,C, D) based on a natural equivalence relation between pairs of linear mappings. We apply this approach to smooth families of quadruples of constant Kronecker type, obtaining smooth families of transformation matrices that pointwise reduce each quadruple to its Kronecker canonical form.

On the geometry of the generalized partial realization problem

Given an observable system (C, A

Feedback reduced and canonical forms for piecewise linear systems

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).

A cellular decomposition of the manifold of observable conditioned invariant subspaces

The geometry of the set of generalized partial realizations of a finite nice sequence of matrices is studied. It is proved that this set is a stratified manifold, the dimension of their strata is computed and its connection with the geometry of the cover problem is clarified. The results can be applied, as a particular case, to the classical partial realization problem.