Further Results on the Decomposition and Serre's Reduction of Linear Functional Systems

Abstract

Given a linear functional system (e.g., ordinary/partial differential system, differential time-delay system, difference system), the decomposition problem aims at studying when it can be decomposed as a direct sum of subsystems. This problem was constructively studied in [4] and the corresponding algorithms were implemented in the OreMorphisms package [5]. Using the OreMorphisms package, many classical linear differential time-delay systems were proved to be decomposable, which highly simplifies the study of their structural properties. Serre's reduction aims at finding an equivalent linear functional system which contains fewer equations and fewer unknowns. It was constructively studied in [1, 6] and successfully applied to different classical examples of differential time-delay systems. Serre's reduction can be seen as a particular case of the decomposition problem. The goal of the present paper is to explicitly provide the links between these two problems. We illustrate the different results with an explicit example of a differential time-delay system.