- $\text{IM}\color{#64B5F6}{\text{PROVING}}$ ON POPOV -

Machine-Certified Absolute Stability for High-Dimensional Lur'e Systems via Lean 4 & Comparator

$Abstract$

High-dimensional Lur'e system stability analysis relies heavily on numerical matrix solvers, leaving safety-critical applications like aerospace EDL vulnerable to frequency-gridding gaps and floating-point drift. To eliminate these approximations, this paper introduces a five-module Popov multiplier framework mechanized via Lean 4 and Comparator. By systematically bridging sector invariants, frequency-domain conditions, and quadratic storage functions through the KYP lemma, we establish an exact, solver-free trajectory stability theorem that guarantees absolute asymptotic convergence across high-dimensional parameter spaces.

✖️ The Popov Multiplier Framework

Before implementing the proofs in Lean 4, the stability problem is structured into five analytical modules:

Module 1 (Plant Definition): Formulate the transfer function $G(s) = C(sI - A)^{-1}B$ over $\mathbb{C}$ and establish its pole stability conditions in Mathlib.

Module 2 (Sector Invariants): Formulate the mathematical properties of sector-bounded functions $\phi \in \text{Sector}[\alpha, \beta]$.

Module 3 (Frequency-Domain Inequalities): State the strict positive real (SPR) condition for the SISO Circle Criterion:

$$\text{Re}\left( \frac{1 + \beta G(j\omega)}{1 + \alpha G(j\omega)} \right) > 0 \quad \forall \omega \in [0, \infty)$$

Module 4 (Popov Criterion Extension): Incorporate the Popov parameter $q \ge 0$ for memoryless time-invariant non-linearities:

$$\text{Re}\left( (1 + j\omega q) G(j\omega) \right) + \frac{1}{k} > 0 \quad \forall \omega \ge 0$$

Module 5 (Lyapunov Equivalence): Connect the frequency-domain SPR condition back to the existence of a quadratic-plus-integral Lyapunov Function $V(x) = x^T P x + q \int_0^y \phi(\sigma) d\sigma$ via the Kalman-Yakubovich-Popov (KYP) Lemma.

✅ Formal Verification in Lean 4 & Comparator Live

The complete mechanical formalization written in Lean 4 implementing the five-module framework and closing the $n$-dimensional absolute stability theorem can be verified and peer reviewed below:

🌐 Verify in Lean Web: 💻 LureSystemAbsoluteStability.lean

🤝 Verify with Comparator Live: 💻 Challenge.lean/Solution.lean

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🏦 Commercial Applications & Industry Use Cases

While this framework was mechanized to eliminate numerical solver approximations in high-dimensional settings, the machine-certified absolute stability proofs have direct safety-critical deployment value:

🚀 Aerospace & Defense (EDL & GN&C): Eliminates frequency-gridding gaps and floating-point drift in spacecraft Entry, Descent, and Landing (EDL) autopilot loops, missile guidance systems, and atmospheric flight controllers where unmodeled resonance spikes or actuator saturation can cause catastrophic divergence.

🤖 Advanced Robotics & Multi-Axis Manipulators: Certifies absolute asymptotic stability for high-degree-of-freedom robotic arms and hydraulic actuation systems operating under strict sector-bounded physical non-linearities and joint constraints.

⚡ Smart Grid & Power Electronics: Guarantees stability certificates for high-capacity inverter-based microgrids and HVDC transmission links where high-frequency switching non-linearities risk destabilizing regional power distribution.

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⚖️ License & Dual-Licensing Policy

This project is open-source software licensed under the GNU Affero General Public License v3.0 (AGPL-3.0).

📜 Commercial Exemption & Proprietary Integration

Due to the strong copyleft provisions of the AGPL, any commercial entity, defense contractor, or enterprise organization that integrates this formal Lean proof, embeds its verification artifacts, or links its formalization into a proprietary, closed-source software product is legally required to make their entire product source code open-source under the terms of the AGPL. For organizations wishing to incorporate these machine-certified stability guarantees into closed-source commercial products or proprietary toolchains without triggering AGPL distribution obligations, commercial exemptions are available.

Disclaimer: Commercial exemptions grant the legal right to bypass AGPL copyleft restrictions for proprietary integration. All formal verification artifacts and proof files are provided "as is", without warranty of any kind, express or implied. The integration, application, validation, and operational safety verification of the code within any commercial product remain entirely the responsibility of the licensee.

💼 To secure a commercial exemption or license, please contact:

Licensing Agent - J.E. Randolph 📧 700josh.r@gmail.com

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📚 Citation

• 📝 Improving on Popov - Machine-Certified Absolute Stability for High-Dimensional Lur'e Systems via Lean 4 & Comparator.pdf

Reed, Jonathan ƒ(n). (2026). Improving on Popov - Machine-Certified Absolute Stability for High-Dimensional Lur'e Systems via Lean 4 & Comparator (Version 1.0). Zenodo. https://doi.org/10.5281/zenodo.22803983

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© 2026 Jonathan ƒ(n) Reed. All rights reserved.