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Jaehong Oh

χPreprint · 2025

Constructive Lyapunov Functions via Topology-Preserving Neural Networks

Jaehong Oh

245 words1 min read

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Abstract

We present a constructive solution to the Lyapunov–Massera–Kurzweil problem via Ontology Neural Networks (ONN), bridging a 60-year gap between existence and construction in stability theory. While Massera (1949) proved that asymptotically stable systems admit Lyapunov functions, his proof was non-constructive, requiring integration over all future trajectories. We demonstrate that the ONN total loss 𝓛_total(S, A) — combining semantic consensus, topological connection, and contextual constraints — serves as an explicit, computable Lyapunov function with closed-form class-𝒦_∞ bounds. Our framework extends classical Lyapunov theory to four challenging domains: (1) non-smooth dynamics via Fejér-monotone topology surgery (60% optimal surgery rate), (2) global stability via persistent homology (Betti-number preservation), (3) delay-differential systems via ORTSF with explicit bounds (τ_max = 177 μs for 3M nodes), and (4) Input-to-State Stability for bounded disturbances. Empirical validation on 3M-node semantic networks demonstrates 99.75% improvement over baseline methods, confirming exponential convergence and topology preservation.

Update (2026-07-10). This 2025 preprint is retained as a manuscript of record for its original framing. Its central claims — that the ONN loss is a constructive Lyapunov function "bridging a 60-year gap", the explicit τ_max = 177 μs for 3M nodes delay bound, and the 99.75% empirical improvement — are not reproducible from the current authoritative research source (onn_ws/ONN) and did not survive the programme's subsequent audit. The honest control result is a standard delay-margin certificate, decoupled from cohomology; the "cohomology as Lyapunov certificate" reading is withdrawn. See the current ONN research status. The abstract below is the manuscript text, unchanged.

Overview

A direct attack on the classical Lyapunov–Massera–Kurzweil problem. Massera's 1949 theorem proved that stable systems have Lyapunov functions, but the proof was non-constructive. This paper shows that the ONN total loss is such a function — explicit, computable, and with closed-form class-𝒦_∞ bounds — closing a 60-year-old gap.

What the paper shows

Four extensions of classical Lyapunov theory:

  1. Non-smooth dynamics via Fejér-monotone topology surgery, with 60% optimal surgery rate.
  2. Global stability via persistent homology (Betti-number preservation).
  3. Delay-differential systems through ORTSF with an explicit bound τ_max = 177 μs for 3M-node systems.
  4. Input-to-State Stability for bounded disturbances.

Empirical validation on 3M-node semantic networks shows 99.75% improvement over baseline methods.

Where it sits

The control-theoretic statement of what the ONN/ORTSF framework buys you: the losses are not just regularisers, they are certificates. Connects directly to the ONN + ORTSF paper.

BibTeX· generated

@misc{oh2025constructive,
  title   = {Constructive Lyapunov Functions via Topology-Preserving Neural Networks},
  author  = {Jaehong Oh},
  year    = {2025},
  url     = {https://jack0682.github.io/papers/constructive-lyapunov-onn/},
}