Extension of the converse Lyapunov proof to nonlinear infinite-dimensional systems

Determine whether the proof technique used to construct implication-form eOSS Lyapunov functions for linear infinite-dimensional systems can be extended to nonlinear infinite-dimensional systems.

Background

The paper proves a converse Lyapunov theorem for eOSS in linear infinite-dimensional systems by constructing a Lyapunov function tailored to regions where the state dominates the output and addressing continuity at the boundary between those regions. The authors note that the finite-dimensional nonlinear proof technique based on local compactness and smooth mollifiers does not transfer directly to infinite-dimensional spaces. Although preliminary lemmas exist, the extension of their own construction to nonlinear infinite-dimensional systems remains unresolved.