Albert Compta

On (A, B)t-invariant subspaces having extendible Brunovsky bases

We consider (A, BY-invariant subspaces having a Brunovsky basis which can be extended to a Brunovsky basis of the whole space. We obtain a geometrical characterization of this class of (A, BY-invariant subspaces, and a complete family of numerical invariants to classify them. 0 Elsevier Science Inc., 1997

Matricial realizations of the solutions of the Carlson problem

Abstract The Klein theorem asserts the existence of at least one solution of the Carlson problem when a related Littlewood–Richardson sequence exists. Then, we present an explicit construction of some matricial solutions.

Miniversal deformations of marked matrices

Abstract Given the set of square matrices M ⊂M n+m (C) that keep the subspace W = C n x {0}⊂ C n + m invariant, we obtain the implicit form of a miniversal deformation of a matrix a∈ M , and we compute it explicitely when this matrix is marked (this is, if there is a permutation matrix p ∈ M n + m ( C ) such that p −1 ap is a Jordan matrix). We derive some applications to tackle the classical Carlson problem.

Dimension of the orbit of marked subspaces

Given a nilpotent endomorphism, we consider the manifold of invariant subspaces having a fixed Segre characteristic. In [Linear Algebra Appl., 332–334 (2001) 569], the implicit form of a miniversal deformation of an invariant subspace with respect to the usual equivalence relation between subspaces is obtained. Here we obtain the explicit form of this deformation when the invariant subspace is marked, and we use it to calculate the dimension of the orbit and in particular to characterize the stable marked subspaces (those with open orbit). Moreover, we study the rank of the endomorphisms in the quotient space by the subspaces in the miniversal deformation of the giving subspace.

The change of feedback invariants under column perturbations: particular cases

We study the variation of the feedback invariants of a complex rectangular n × (n + m) matrix when we make small additive perturbations to the elements of the last m columns. First of all, we obtain necessary conditions for the feedback invariants of all the matrices obtained by means of sufficiently small perturbations. Conversely, we prove that these conditions are also sufficient to find a matrix, as close as desired to the fixed matrix, with prescribed feedback invariants, in some particular cases: when the rectangular matrix is completely controllable, when the rectangular matrix is completely uncontrollable and when m = 1.

Local differentiable pole assignment

Given a general local differentiable family of pairs of matrices, we obtain a local differentiable family of feedbacks solving the pole assignment problem, that is to say, shifting the spectrum into a prefixed one. We point out that no additional hypothesis is needed. In fact, simple approaches work in particular cases (controllable pairs, constancy of the dimension of the controllable subspace, and so on). Here the general case is proved by means of Arnolds techniques: the key point is to reduce the construction to a versal deformation of the central pair; in fact to a quite singular miniversal one for which the family of feedbacks can be explicitly constructed. As a direct application, a differentiable family of stabilizing feedbacks is obtained, provided that the central pair is stabilizable.

Geometric classification of monogenic subspaces and uniparametric linear control systems

We present a geometric approach to the classification of monogenic invariant subspaces, alternative to the classical algebraic one, which allows us to obtain several matricial canonical forms for each class. Some applications are derived: canonical coordinates of a vector with regard to an endomorphism, and a canonical form for uniparametric linear control systems, not necessarily controllable, with regard to linear changes of state variables. Moreover, the pointwise construction can be extended to differentiable families of changes of basis when differentiable families of equivalent monogenic subspaces are considered.

Miniversal deformations of observable marked matrices

Given the set of vertical pairs of matrices

GEOMETRIC STRUCTURE OF SINGLE/COMBINED EQUIVALENCE CLASSES OF A CONTROLLABLE PAIR ∗

{\cal M}\subset M_{m,n}(\mathbb C)\times M_n(\mathbb C)

Output Regulation Problem for Differentiable Families of Linear Systems

keeping the subspace