Connections between integral quadratic constraints and dissipativity

We show that a recent dissipativity approach to feedback stability analysis of potentially open-loop unstable systems, which encompasses the classical soft integral quadratic constraint (IQC) theorem, may be recovered by hard IQC theory. The latter is known to be subsumable by the more general soft IQC theory endowed with homotopies that are continuous in the gap topology. Additionally, we demonstrate how the aforementioned classical soft IQC theorem, initially introduced for the analysis of a feedback interconnection of a nonlinear component and a linear system, may be recast to analyse the stability of a feedback interconnection of two nonlinear systems. This generates a frequency-dependent $(Q(\omega), S(\omega), R(\omega))$ -dissipativity result.