Systems Theory over Finitely Generated Abelian Groups

Abstract

A decision which has to be made in systems theory regards the choice of the space where the signals are assumed to take place. This is a crucial choice when we deal with continuous-time systems governed by non-linear or by partial differential equations while it becomes a less relevant issue when we are dealing with linear systems described by linear ordinary differential equations. In this paper we will instead focus our attention on discrete-time systems. In the linear case, the standard choice made in the classical set up are the Laurent sequence spaces for input and output signals [8] (namely signals are supposed to be 0 in the far past). On the other hand, the behavioral approach proposed by Willems [12, 13, 14] tends to privilege the complete sequence spaces (no constraints on the signals neither in the past nor in the future and so no time asymmetry). The difference, in the linear case, consists in adding autonomous phenomena which are absent when we work with the Laurent sequence spaces, while the controllable systems can be equivalently represented in the both frameworks. The situation is different when we abandon the linear setting and we consider more general systems over groups and rings for which there has been recently a lot of interest for their applications in coding [6, 10, 11, 3].