Lyapunov criteria for boundedness of reachability sets of distributed parameter systems

Andrii Mironchenko

correctmedium confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves full equivalences for both BRS (Theorem 3.5) and RFC (Theorem 4.3), including the converse Lyapunov directions. In particular, for BRS the paper explicitly establishes (iv) ⇒ (v) via a constructed Lipschitz (on bounded balls) Lyapunov function W and (v) ⇒ (i) with a detailed argument using BIC and cocycle properties, thereby showing (i)–(v) are equivalent; see the statement of Theorem 3.5 and its proof steps including the Vk and W constructions and the derivation of bound (10) . The model incorrectly asserts that the BRS converse (i)⇒(v) was likely open; the paper resolves it. For RFC, the model’s equivalences align with Theorem 4.3, which states the same (i)–(v) equivalences under BIC and uniform Lipschitz flow on compact intervals over D .

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript establishes clean, constructive Lyapunov characterizations of BRS and RFC in an abstract setting (BIC, Lipschitz flows on compact intervals). It generalizes finite-dimensional results, addresses infinite-dimensional subtleties, and provides a converse Lyapunov construction. Proofs are sound and readable. Minor editorial clarifications (on Lipschitz regularity phrasing and a brief sketch of standard equivalences) would enhance clarity.