The pointwise feedback relation for linear dynamical systems Original Research Article

Let ∑ and ∑ʹ be two m-input n-dimensional linear dynamical systems over a commutative ring R. If ∑ is feedback equivalent to ∑ʹ then ∑ is pointwise feedback equivalent to ∑ʹ (i.e. for all prime ideal p of R the systems ∑(p) and ∑ʹ(p) are feedback equivalent over the residue field k(p), where ∑(p) is the natural extension of ∑ to k(p)). This paper is devoted to the study of the pointwise feedback relation. We characterize when two systems ∑ and ∑ʹ are pointwise equivalent. We show that R is an absolutely flat ring if and only if the feedback relation and the pointwise feedback relation are equivalent. The particular case of rings of real-valued continuous functions is specially treated.