Topological aspects of the partial realization problem

On a cell decomposition for Hankel matrices and rational functions.

“…In a different context we derived in [22] a stratification of the space Hank(«, n χ n) of nonsingular real or complex n x n Hankel matrices from the Bruhat decomposition of the general linear group G1"(W), IK= R or C. In the real case the Bruhat strata are a collection of disjoint cells. Applications to some topological aspects of the partial realization problem can be found in [32]. Using this correspondence we proved a weak version of the conjecture of Fuhrmann and Krishnaprasad, namely that the continued fraction cells form a cell decomposition of Rat (n) in the sense of Definition 2.1.…”

On the geometry of the generalized partial realization problem

“…It is not surprising then that since the first treatment of this problem by Kalman [21,22], many authors have investigated various aspects of the problem such as parametrizations, algorithms for constructing realizations, extensions to linear descriptor systems, topological and geometric aspects of the problem, etc. (see among others [4,5,16,19,23,25,34]). …”

“…, h n−1 ) which admit a minimal realization of fixed order and the other about the geometry of the set of minimal systems that realize a given such a sequence. Brockett himself provided a first approach to the solution of the former problem and several years later the task was completed in [25]. The aim of the present paper is to provide with a differentiable structure the set of solutions of the generalized partial realization problem; a big generalization of Brockett's second problem (see below).…”