A Geometric Approach to the Carlson Problem

Abstract

Abstract. The possible observability indices of an observable pair of matrices, when supplementary subpairs are prescribed, are characterized when the "quotient" one is nilpotent. The geometric techniques used are also valid in the classical Carlson problem for square matrices.

Geometric classification of monogenic subspaces and uniparametric linear control systems

“…We recall that it asks for conditions to ensure when three given Weyr characteristics can be realized as the ones corresponding to an endomorphism, its restriction to an invariant subspace and the one induced in the quotient space. The LR-sequences give an implicit answer to this problem ( [12], [2]), but they do not characterize the equivalence class of a general invariant subspace, whereas we show that it is so for monogenic ones. Indeed, we find (Corollary 3.9) families of classifying elements, alternative to the Ulm sequence in [11], which can be easily listed (Corollary 6.8).…”

“…For monogenic subspaces, the classifying geometric parameters are the so-called marked and perturbation indices defined by means of the LR-sequence associated to each invariant subspace in [2] (Definition 3.4). The key tool in this paper is the geometrical approach there to the Carlson problem.…”

“…In Section 2 we recall the basic definitions and results concerning invariant subspaces which will be used in the sequel. In particular, the Carlson problem, the LR-sequences and the techniques in [2] based in the double Jordan filtration which will be the key tool in our reasonings. In Section 3 we focus in the monogenic case.…”

Miniversal deformations of observable marked matrices

“…In , the ‘interesting class’ of the so‐called ‘marked’ subspaces, namely the invariant subspaces having a Jordan basis that can be extended to a Jordan basis of the whole space, is introduced. For instance, in , one proves that the ‘simplest’ solutions of the Carlson problem are marked, and any other solution appears in a neighborhood of the marked ones. This notion was extended to pairs of matrices in and used in for the analog to the Carlson problem: Again, the marked solutions cover all the possibilities and are the simplest realizations.…”

Perturbed marked reduced forms of invariant subspaces

“…We denote by Inv(p, q) the set of these invariant pairs (in [13] one shows that it is a differentiable manifold). Moreover, if two invariant pairs are equivalent then also the quotient endomorphisms must have the same Segre characteristic, but there are no explicit criteria in order to determine if a quotient Segre characteristic is compatible with p and q (the Carlson problem: see [2], [3]).…”