Equivalences of Linear Functional Systems

Abstract

Within the algebraic analysis approach to linear systems theory, we investigate the equivalence problem of linear functional systems, i.e., the problem of characterizing when all the solutions of two linear functional systems are in a one-to-one correspondence. To do that, we first provide a new characterization of isomorphic finitely presented modules in terms of inflations of their presentation matrices. We then prove several isomorphisms which are consequences of the unimodular completion problem. We then use these isomorphisms to complete and refine existing results concerning Serre's reduction problem. Finally, different consequences of these results are given. All the results obtained in this paper are algorithmic for rings for which Gröbner basis techniques exist and the computations can be performed by the Maple packages OREMODULES and OREMORPHISMS.

Effective Algebraic Analysis Approach to Linear Systems over Ore Algebras

“…We also note that hom i.e., there is a 1-1 correspondence between the solutions of the first system and the solutions of the second one. For more details and applications of this result to Serre's reduction, Stafford's reduction, the decomposition problem, see [12,14,47] and the references therein. Two different representations R ∈ D q×p and R ∈ D q ×p of the same linear system define two isomorphic modules:…”

“…Let O := B[∂;id A , δ] be the skew polynomial ring of OD operators with coefficients in B := A or K. The generic linearization of (14) is then defined by R η = 0, where R := ∂ I n − ∂f ∂x − ∂f ∂u ∈ D n×(n+m) and η := (dx T du T ) T , and can be studied by means of the finitely presented left O-module M := O 1×(n+m) /(O 1×n R). The cases of a rational, analytic or meromorphic function f can be studied similarly by considering the differential ring or field B formed by the rational/analytic/meromorphic functions which satisfy (14).…”

“…Within the algebraic analysis approach to linear systems theory [9,43,47,51,57] For the details of these results, see the chapter [14] of this book. Note that the y j 's do not belong to F but are just elements of M .…”

“…For a direct proof of Theorem 3, see the chapter [14] of this book. Using the isomorphism (15), the linear system ker F (R.) depends only on M and F. Hence, we can study its built-in properties by means of those of the modules M and F. Note that the functional space F where the solutions are sought can be altered and the behaviour of the solutions highly depend on it (in a similar way as for the F-solutions of x 2 + 1 = 0 for F := R or C) [43].…”