Noncoherence of a Causal Wiener Algebra Used in Control Theory

Abstract

Let ${\mathbb{C}}_{\geq 0}:=\{s\in \mathbb{C}\mid \text{Re}(s)\geq 0\}$, and let ${\mathcal{W}}^{+}$ denote the ring of all functions $f:{\mathbb{C}}_{\geq 0} \rightarrow \mathbb{C}$ such that $f(s)={f}_{a}(s)+\displaystyle{{\sum }_{k=0}^{\infty }{f}_{k}{e}^{-s{t}_{k}}}\,(s\in {\mathbb{C}}_{\geq 0})$, where ${f}_{a}\in {L}^{1}(0,\infty ),\,{({f}_{k})}_{k\geq 0}\in {\ell }^{1}$, and $0={t}_{0}< {t}_{1}< {t}_{2}< \cdots $ equipped with pointwise operations. (Here $\widehat{{\cdot}}$ denotes the Laplace transform.) It is shown that the ring ${\mathcal{W}}^{+}$ is not coherent, answering a question of Alban Quadrat. In fact, we present two principal ideals in the domain ${\mathcal{W}}^{+}$ whose intersection is not finitely generated.