Subsistemas singulares de un sistema lineal. Una aproximación a los subespacios cuasiinvariantes

Dado un sistema lineal ˙x = Ax + By, se definen los subespacios cuasi-(A, B)- invariantes como aquellos tales que para cada condición inicial en el subespacio, existe un control u(t) que hace que la correspondiente trayectoria, pertenezca enteramente en un entorno arbitrariamente próximo al subespacio. En este trabajo caracterizamos los subespacios cuasi-(A, B)-invariantes a través de existencia de un sistema singular, que puede interpretarse como la restricción del sistema definido por (A, B) al subespacio, pero en el cual se admiten distribuciones como ‘funciones’ de control

Classification and versal deformation of generalized flags

A natural equivalence relation can be considered in the generalized flag manifold. First we give a complete set of invariants of it as well as a canonical matrix description of the classes. Next we consider parametric flags. We give a miniversal deformation for the above canonical form and we use it to characterize the stable flags.Peer Reviewe

Versal deformations in generalized flag manifolds

We find explicit miniversal deformations of flags in the generalized flag manifolds, with regard a natural equivalence relation defined by the group action that keeps fixed the reference flag

On the perturbation of bimodal systems

Given a bimodal system de¯ned by the equations ½ x_ (t) = A1x(t) + Bu(t) if ctx(t) · 0 x_ (t) = A2x(t) + Bu(t) if ctx(t) ¸ 0 (1) where B 2Mn;m and Ai 2Mn, i = 1; 2, are such that A1;A2 coincide on the hyper- plane V =Kerct. We consider in the set of matrices de¯ning the above systems the simultaneous feedback equivalence de¯ned by ([A1;B]; [A2;B]) » ([A0 1;B0]; [A0 2;B0]) if [A0 i B0] = S¡1[Ai B] · S 0 R T ¸ i = 1; 2 with S(V) = V This equivalent relation corresponds to the action of a Lie group. Under this action we obtain, in the case m · 1, the semiuniversal deformation, following Arnold's technique. Then the problem of structural stability is studied.Postprint (published version

On the geometry of the solutions of the cover problem

For a given system (A;B) and a subspace S, the Cover Problem consits of ¯nding all (A;B)-invariant subspaces containing S. For controllable systems, the set of these subspaces can be suitably strati¯ed. In this paper, necessary and su±cient conditions are given for the cover problem to have a solution on a given strata. Then the geometry of these solutions is studied. In particular, the set of the solutions is provided with a di®erentiable structure and a parametrization of all solutions is obtained through a coordinate atlas of the corresponding smooth manifold

On the parametrization of the controllability subspaces of a controllable pair

Given a controllable linear control system defined by a pair of constant matrices (A;B), the set of controllability subspaces is an stratified submanifold of the set of (A;B)-invariant subspaces. We parameterize each strata by means of coordinate charts. This parametrization has significant di®erences to that of (A;B) invariant subspaces, showing a more complex geometric structure

Stratification and bundle structure of the set of general (A,B)-invariant subspaces

Given (A,B) in Hom(C^(n+m) C^n), we prove that the set of (A,B)-invariant subspaces having a fixed Brunovsky-Kronecker structure is a submanifold of the corresponding grassman manifold, and we compute its dimension. Also, we prove that the set of all (A,B)-invariant subspaces having a fixed dimension is connected, provided that (A,B) has only one eigenvalue

Stability of (A,B)-invariant subspaces

Given a pair of matrices (A;B) we study the stability of their invariant subspaces from the geometry of the manifold of quadruples (A;B; S; F) where S is an (A;B)-invariant subspace and F is such that (A + BF)S ½ S. In particular, we derive a su±cient computable condition of stability

A sufficient condition for Lipschitz stability of controlled Invariant subspaces

Given a pair of matrices (A,B) we study the Lipschitz stability of its controlled invariant subspaces. A sufficient condition is derived from the geometry of the set formed by the quadruples (A,B, F, S) where S is an (A,B)-invariant subspace and F a corresponding feedback.Peer ReviewedPostprint (published version

Simultaneous versal deformations of endomorphisms and invariant subspaces

We study the set M of pairs (f; V ), defined by an endomorphism f of Fn and a d- dimensional f–invariant subspace V . It is shown that this set is a smooth manifold that defines a vector bundle on the Grassmann manifold. We apply this study to derive conditions for the Lipschitz stability of invariant subspaces and determine versal deformations of the elements of M with respect to a natural equivalence relation introduced on it