Noncoherence of a Causal Wiener Algebra Used in Control Theory

Amol Sasane

Abstract and Applied Analysis, 2008, vol. 2008, 1-13

Abstract:

Let â„‚ ≥ 0 ∶ = { ð ‘ âˆˆ â„‚ ∣ R e ( ð ‘ ) ≥ 0 } , and let ð ’² + denote the ring of all functions ð ‘“ ∶ â„‚ ≥ 0 → â„‚ such that ð ‘“ ( ð ‘ ) = ð ‘“ ð ‘Ž ( ð ‘ ) + ∑ ∞ 𠑘 = 0 ð ‘“ 𠑘 ð ‘’ − ð ‘ ð ‘¡ 𠑘 ( ð ‘ âˆˆ â„‚ ≥ 0 ) , where ð ‘“ ð ‘Ž ∈ ð ¿ 1 ( 0 , ∞ ) , ( ð ‘“ 𠑘 ) 𠑘 ≥ 0 ∈ â„“ 1 , and 0 = ð ‘¡ 0 < ð ‘¡ 1 < ð ‘¡ 2 < ⋯ equipped with pointwise operations. (Here 0 ð ‘¥ 0 0 0 5 ð ‘’ â‹… denotes the Laplace transform.) It is shown that the ring ð ’² + is not coherent, answering a question of Alban Quadrat. In fact, we present two principal ideals in the domain ð ’² + whose intersection is not finitely generated.

Date: 2008
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:459310

DOI: 10.1155/2008/459310

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