Systems over Rings: Geometric Theory and Applications

The aim of this paper is to present an overview of the geometric approach to the study of dynamical systems with coefficients in a ring and, in particular, to discuss the most recent results concerning noninteracting control problems it has produced. The study of systems with coefficients in a ring is motivated, in addition to an intrinsic interest on the subject, by the fact that they may be viewed as tools for investigating problems which concern parameter dependent systems or delay-differential systems. The geometric approach is shown to provide complete characterizations of the solvability conditions, together with feasible procedures for computing solutions, of noninteracting control problems, like the Disturbance Decoupling Problem and the Block Decoupling Problem. Finally, some examples which clarify the applications to problems concerning delay-differential systems are presented.