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

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.