Learning Koopman-based Stability Certificates for Unknown Nonlinear Systems

The paper’s Proposition 4.2 establishes a uniform model-error bound (its eqs. (10)–(11)) and then shows strict negativity of the true Lie derivative on the annulus Ω_{c1,c2}, concluding that trajectories from Ω_{c2} converge into Ω_{c1} and then to the origin under Assumption 4.1, provided Ω_{c2} does not intersect ∂X. The candidate solution reproduces the same bound and negativity margin, but makes explicit the forward-invariance of {V ≤ c2} and the finite-time entrance to {V ≤ c1}. These additions are consistent with and strengthen the paper’s concise proof. No contradictions were found. Key ingredients and assumptions (local Lipschitz dynamics, global existence I = [0,∞), Assumption 4.1, and the margin condition) match the paper’s setup and proof of Prop. 4.2.