A homotopy theorem for incremental stability

TL;DR

The paper introduces an incremental homotopy framework for proving incremental stability in feedback interconnections, replacing gap-metric perturbations with incremental gain bounds. It derives two corollaries: a separation criterion based on Scaled Relative Graphs (SRG) to verify incremental stability and an incremental version of the IQC stability theorem, applicable beyond linear time-invariant systems. The approach does not require extended spaces or causality assumptions and allows relaxation of well-posedness prerequisites in the incremental setting, extending applicability to nonlinear operators. Together, these results provide a practical, generalizable toolkit for incremental robustness analysis and connect to related SRG and differential IQC developments.

Abstract

A theorem is proved to verify incremental stability of a feedback system via a homotopy from a known incrementally stable system. A first corollary of that result is that incremental stability may be verified by separation of Scaled Relative Graphs, correcting two assumptions in [1, Theorem 2]. A second corollary provides an incremental version of the classical IQC stability theorem.