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.
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.
Four extensions of classical Lyapunov theory:
- Non-smooth dynamics via Fejér-monotone topology surgery, with
60% optimal surgery rate.
- Global stability via persistent homology (Betti-number
preservation).
- Delay-differential systems through ORTSF with an explicit
bound τ_max = 177 μs for 3M-node systems.
- Input-to-State Stability for bounded disturbances.
Empirical validation on 3M-node semantic networks shows 99.75%
improvement over baseline methods.
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.