Local Wiener--Hopf factorization and indices over arbitrary fields
Comprising four sections and two appendices, the paper gives generalizations to arbitrary fields of the usual Wiener--Hopf equivalence of complex valued rational matrix functions. Also, the local Wiener--Hopf factorization indices introduced in [Linear Algebra Appl.\ 425, No~2--3, 757--775 (2007; Zbl 1126.93020)] are proved to form a complete system of invariants for this equivalence relation. The first two sections present a detailed history of the topic and introduce some necessary definitions. In the third section, the local Wiener--Hopf equivalence of non-singular rational matrices over an arbitrary field with respect with two subsets in the set of all maximal ideals of the ring of polynomials with coefficients in the considered field is defined. The main results are that the factorization indices form a complete system of invariants, and, under certain conditions, every non-singular polynomial matrix is equivalent to a diagonal matrix. In the fourth section, the obtained results are applied to linear control systems. Also is proved here that any controllable pair can be reduced by feedback equivalence to a reduced form similar to the Brunovski canonical form. The paper concludes with two appendices which present the basic Wiener--Hopf factorization theorem over the complex field, and the generalization procedure to arbitrary fields. Also, the left Wiener--Hopf equivalence of rational matrix functions is analysed
scientific article; zbMATH DE number 5518782
scientific article