The pointwise feedback relation for linear dynamical systems
\(R\) denotes a commutative ring. If \(\Sigma\) is an \(m\)-input \(n\)-dimensional linear dynamical system and \(\mathbf p\) is a prime ideal, then \(\Sigma (\mathbf p)\) denotes the natural extension of \(\Sigma\) to \(k(\mathbf p),\) the residual field of \(R\) at \(\mathbf p\). Two systems \(\Sigma\) and \(\Sigma '\) over \(R\) are called pointwise feedback equivalent if \(\Sigma (\mathbf p)\) and \(\Sigma ' (\mathbf p)\) are feedback equivalent over \(k(\mathbf p)\) for all prime ideals \(\mathbf p\) of \(R\). A set of invariants are constructed for the equivalence class of \(\Sigma\) by the action of the feedback group. An example is given to show that these invariants are not sufficient to specify the feedback equivalence class of \(\Sigma\) in general, but it is proved that the feedback class of a reachable linear dynamical system over a field is characterized by the constructed invariants. In fact, the set of Kronecker indices may be deduced from these invariants in this case. Turning to pointwise feedback equivalence, a characterization is found in terms of the set of the constructed invariants and proved to be of topological nature. The relationship between feedback equivalence and pointwise feedback equivalence is then investigated. The main result states that a ring \(R\) is absolutely flat if and only if the feedback relation and the pointwise feedback relation are equivalent.
scientific article; zbMATH DE number 1355343