Noncoherence of a Causal Wiener Algebra Used in Control Theory

Abstract

Let ${ℂ}_{\ge 0}:=\{s\in ℂ\mid \text{Re}(s)\ge 0\}$, and let ${𝒲}^{+}$ denote the ring of all functions $f:{ℂ}_{\ge 0}\to ℂ$ such that $f(s)=f_a(s)+{\displaystyle {\sum }_{k=0}^{\infty }{f}_k{e}^{-s{t}_k}}\hspace{0.17em}(s\in {ℂ}_{\ge 0})$, where $f_a\in L^1\left(0,\infty \right),\hspace{0.17em}{\left(f_k\right)}_{k\ge 0}\in {\ell }^1$, and equipped with pointwise operations. (Here $\widehat{\cdot }$ denotes the Laplace transform.) It is shown that the ring ${𝒲}^{+}$ is not coherent, answering a question of Alban Quadrat. In fact, we present two principal ideals in the domain ${𝒲}^{+}$ whose intersection is not finitely generated.