The new $\nu$-metric induces the classical gap topology

TL;DR

This paper demonstrates that a newly introduced $ u$-metric for unstable plants over a specific function space aligns with the classical gap topology, providing a unified framework for robust stabilization analysis.

Contribution

It shows that the extended $ u$-metric induces a topology equivalent to the classical gap topology on unstable plants over $\\calA_+$, bridging two important concepts.

Findings

The new $ u$-metric coincides with the classical gap topology.

The topology induced by the new $ u$-metric is equivalent to the gap topology.

This equivalence facilitates robust stabilization analysis.

Abstract

Let $\calA_+$ denote the set of Laplace transforms of complex Borel measures $\mu$ on $[0,+\infty)$ such that $\mu$ does not have a singular non-atomic part. In \cite{BalSas}, an extension of the classical $\nu$-metric of Vinnicombe was given, which allowed one to address robust stabilization problems for unstable plants over $\calA_+$. In this article, we show that this new $\nu$-metric gives a topology on unstable plants which coincides with the classical gap topology for unstable plants over $\calA_+$ with a single input and a single output.