Stability Analysis and Data-driven Verification via Recurrent Lyapunov Functions

Roy Siegelmann; Fernando Paganini; Enrique Mallada

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arXiv · 2608.26447

Stability Analysis and Data-driven Verification via Recurrent Lyapunov Functions

Roy Siegelmann·Fernando Paganini·Enrique Mallada

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Abstract

Lyapunov's direct method is a cornerstone of stability and control, but it hinges on finding a Lyapunov function, a task demanding ingenuity or computation. A key difficulty is that every sub-level set of the function must be forward invariant, coupling its geometry to the system's trajectories. We relax this by replacing invariance with recurrence: a set is recurrent if every trajectory starting in it returns within a finite time. This yields the notion of a Recurrent Lyapunov Function (RLF), whose sub-level sets need only be recurrent. We show that, under mild conditions, RLFs guarantee stability, and we introduce stronger notions yielding asymptotic and exponential stability. We also give norm-based converse theorems: under the corresponding stability conditions, any norm is an RLF for their practical versions. We then develop GPU-based algorithms that certify (practical) stability from trajectory data alone, without a Lyapunov function. Certifying stability up to an $\varepsilon$-neighborhood needs only $O(\log(1/\varepsilon))$ trajectory evaluations, with constants growing as the certified decay rate nears the true one, exposing an intrinsic performance-cost trade-off.

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Roy Siegelmann, Fernando Paganini, Enrique Mallada. 2026-08-26. Stability Analysis and Data-driven Verification via Recurrent Lyapunov Functions. https://arxiv.org/abs/2608.26447

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