Existence of stable Lur'e systems for which the O'Shea-Zames-Falb stability test fails

What the math gives to ML

The paper demonstrates that the restrictive O'Shea-Zames-Falb multiplier family can fail even when a Lur'e interconnection is robustly stable, while a more general full-block IQC multiplier proves stability. The transferable asset is the combination of finite-horizon lifted nonlinearities and a strict semidefinite feasibility certificate. This suggests replacing heuristic spectral-norm constraints for recurrent or residual networks with optimized IQC certificates that directly account for activation slope restrictions and linear state dynamics. A practical first version can search for a multiplier and Lyapunov matrix offline, then use the resulting LMI margin as a stability regularizer or architecture filter.