On stability analysis with the v-gap metric and integral quadratic constraints

This is part of an effort to extend Vinnicombe´s v-gap metric based analysis of uncertain feedback interconnections to a linear time-varying setting. The first results involved using integral quadratic constraints (IQCs) to characterise the uncertainty and the existence of v-gap homotopies. Recent work establishes more familiar results in terms of v-gap balls, via the properties of graph symbols and generalised Wiener-Hopf and Hankel operators revealed by the initial work. In this paper, the additional flexibility of IQC based analysis is reconciled with a v-gap ball based stability result. That is to say, we show the latter can be recovered within the original IQC and v-gap homotopy based framework. To this end, path-connectedness of v-gap balls plays a central role. This is established by exploiting a linear fractional characterisation of the v-gap metric and the existence of a certain J-spectral factorisation, which is shown to be the case for finite-dimensional systems with stabilisable and detectable state-space realisation.