Brunovsky's canonical form for linear dynamical systems over commutative rings
\textit{P. A. Brunovsky} [Kybernetika 3, 173-188 (1970; Zbl 0199.48202)]\ showed that a reachable linear dynamical system \((F, G)\) over a field \(k\) is feedback equivalent to a system \((F_c G_c)\) which has a canonical form determined completely by invariants \(k_1, \dots, k_s\), called the Kronecker indices. Examples showing that Brunovsky's classification is not valid for systems over a general commutative ring have long been known. Systems over a commutative ring \(R\) with unit that are feedback equivalent to a system \((F_c, G_c)\) (as for a field), are called Brunovsky systems here. In the main theorem of this paper, the authors succeed in characterizing the Brunovsky systems over a commutative ring \(R\) with unit such that finitely generated projective \(R\)-modules are free. The characterization is obtained in terms of the minors of the matrices \((G, FG, \dots, F^{i-1} G)\) for \(1\leq i\leq n\) and a complete set of invariants which generalizes the Kronecker indices is obtained. A determinantal method to obtain this set of invariants is included as well, yielding an alternative method to obtain the Kronecker indices in the case of a system over a field.
scientific article; zbMATH DE number 841742
scientific article