Time-varying generalisations of the gap and ν-gap metrics induce the same topology in continuous time

Recent robust stability analysis results for linear time-varying feedback interconnections are based on a time-varying generalisation of the ν-gap metric. The causality of closed-loop mappings is dealt with explicitly, rather than via well-posedness assumptions as is common in the literature. Here, an alternative time-varying gap metric is defined. It is shown that this gives rise to corresponding robust stability results. It is also established that the time-varying gap metric induces the same topology as the generalised ν-gap metric, this being the coarsest under which closed-loop stability and performance are both robust properties.