Isomorphisms and Serre’s reduction of linear functional systems

Within the algebraic analysis approach to linear systems theory, a behaviour is the dual of the left module finitely presented by the matrix of functional operators defining the linear functional system. In this talk, we give an explicit characterization of isomorphic finitely presented modules, i.e., of isomorphic behaviours, in terms of certain inflations of their presentation matrices. Fitting’s theorem (see [3] and references therein) on the syzygy modules can be found again. If one of the presentation matrix has full row rank, this result yields a characterization of isomorphic modules as the completion problem characterizing Serre’s reduction, i.e., the possibility to find a presentation of the module defined by fewer generators and fewer relations, and thus an equivalent representation of the behaviour defined by fewer equations in fewer unknown functions (see [1] and references therein). This completion problem is shown to induce different isomorphisms between the modules finitely presented by the matrices defining the inflations. Applications to doubly coprime factorizations are given. Finally, we will show that Serre’s reduction implies the existence of a certain idempotent endomorphism of the finitely presented module, i.e., a particular decomposition problem (see [2]), proving the converse of a result obtained in [4].