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arxiv: 2605.19961 · v2 · pith:T6NX3LUSnew · submitted 2026-05-19 · 🧮 math.OC · cs.SY· eess.SY

Data-driven approximation of regions of attraction via an LP-based selection of PWA Lyapunov functions

Oumayma Khattabi , Matteo Tacchi-B\'enard , Martin Gulan , Sorin Olaru This is my paper

Pith reviewed 2026-05-21 07:39 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords data-driven stabilityregion of attractionpiecewise affine Lyapunovlinear programminguncertainty setnonlinear dynamicsrobust decreasestability certification
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The pith

A linear program synthesizes a continuous piecewise affine Lyapunov function to certify a region of attraction consistent with pointwise data and Lipschitz bounds for unknown nonlinear systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a method to approximate regions of attraction for nonlinear dynamical systems when only pointwise evaluations of the vector field and Lipschitz bounds are known. It constructs a polyhedral uncertainty set that contains all possible dynamics consistent with this information. A linear program is then used to synthesize a continuous piecewise affine Lyapunov function that decreases for every vector field in the uncertainty set. If successful, this certifies that trajectories starting in a certain region will converge to the equilibrium based solely on the available data. This matters for systems where a complete model is unavailable but some measurements exist.

Core claim

By constructing a polyhedral uncertainty set encompassing all admissible vector fields from point-wise data and Lipschitz constants, the method formulates a linear program to compute a continuous piecewise affine Lyapunov candidate that satisfies a robust decrease condition across the entire set, thereby certifying a data-consistent region of attraction.

What carries the argument

The linear program that optimizes the parameters of a continuous piecewise affine Lyapunov function subject to robust decrease inequalities over the polyhedral uncertainty set of dynamics.

If this is right

  • The certified region is guaranteed to be invariant and attracting for any dynamics inside the uncertainty set.
  • Stability certificates can be extracted directly from experimental data without assuming a specific model form.
  • The approach works with sparse data points provided the Lipschitz bounds are supplied.
  • Numerical examples demonstrate extraction of certified regions even when information is limited.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Iteratively adding more data points could refine the uncertainty set and enlarge the certified region over time.
  • The same LP framework might be extended to design feedback laws that maximize the size of the certified region.
  • Computational cost grows with the number of facets in the polytope, pointing to a need for adaptive partitioning in higher dimensions.

Load-bearing premise

The polyhedral uncertainty set constructed from the point-wise evaluations and known Lipschitz bounds is assumed to contain every vector field consistent with the data.

What would settle it

Discovery of a vector field that agrees with the data at the evaluation points, respects the Lipschitz bound, but has at least one trajectory starting inside the certified region that diverges from the origin.

Figures

Figures reproduced from arXiv: 2605.19961 by Martin Gulan, Matteo Tacchi-B\'enard, Oumayma Khattabi, Sorin Olaru.

Figure 1
Figure 1. Figure 1: Illustration of Pi,κ with K ∈ {1, 2} (in green, yellow, orange and red are P (+) i,κ,1 , P (−) i,κ,1 , P (+) i,κ,2 and P (−) i,κ,2 respectively). -1 1 -0.5 0.5 1 0 0.5 0 0.5 0 1 -0.5 -0.5 -1 -1 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of Qκ with 4 data points. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Dynamics of a simple damped pendulum [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Lyapunov candidate and RoA for the damped pendulum. [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Dynamics of an inverted-time Van der Pol oscillator. [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Lyapunov candidate and RoA for the Van der Pol oscillator. [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
read the original abstract

This paper presents a method to approximate regions of attraction of unknown nonlinear dynamical systems from data. Assuming point-wise evaluations of the vector field and known Lipschitz bounds, a polyhedral uncertainty set of admissible dynamics is constructed. This uncertainty description enables the synthesis of a continuous piece-wise affine Lyapunov candidate via a linear program, enforcing a robust decrease condition for all admissible vector fields. The approach allows certification of a region of attraction consistent with the available data. Numerical examples illustrate the effectiveness of the proposed method in extracting certified regions of attraction from sparse data.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper presents a method to approximate regions of attraction of unknown nonlinear dynamical systems from data. Assuming point-wise evaluations of the vector field and known Lipschitz bounds, a polyhedral uncertainty set of admissible dynamics is constructed. This uncertainty description enables the synthesis of a continuous piece-wise affine Lyapunov candidate via a linear program, enforcing a robust decrease condition for all admissible vector fields. The approach allows certification of a region of attraction consistent with the available data. Numerical examples illustrate the effectiveness of the proposed method in extracting certified regions of attraction from sparse data.

Significance. If the central claims hold, the work provides a computationally tractable LP-based route to certified RoA estimates from limited data and Lipschitz information, which could be useful for verification in data-driven control. The combination of polyhedral uncertainty modeling with continuous PWA Lyapunov functions is a reasonable technical direction, though its practical impact depends on resolving the inclusion and robustness questions below.

major comments (2)
  1. [Section 3.2] Section 3.2: The polyhedral uncertainty set is constructed from point-wise evaluations plus Lipschitz bounds, but the manuscript does not demonstrate that the facet inequalities exactly contain every vector field consistent with the data and the Lipschitz condition. Without this inclusion guarantee, the robust decrease condition enforced by the LP cannot be guaranteed to hold for all admissible dynamics, directly undermining the certification claim.
  2. [Section 4] Section 4: The LP formulation for selecting the PWA Lyapunov function is described at a high level, but it is not shown how the robust decrease condition over the entire polyhedral set is reduced to a finite number of linear constraints (e.g., via vertex or facet enumeration). This step is load-bearing for the claim that the synthesized function certifies a valid RoA.
minor comments (2)
  1. [Numerical examples] The choice of state-space partition for the PWA function is listed as a free parameter; a brief discussion of how it is selected in the numerical examples would improve reproducibility.
  2. [Figures] Figure captions could more explicitly indicate which trajectories or sets correspond to the certified RoA versus the uncertainty bounds.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments on our manuscript. We believe the suggested clarifications will improve the rigor of the presentation. We address each major comment below and plan to incorporate the necessary revisions.

read point-by-point responses

  1. Referee: Section 3.2: The polyhedral uncertainty set is constructed from point-wise evaluations plus Lipschitz bounds, but the manuscript does not demonstrate that the facet inequalities exactly contain every vector field consistent with the data and the Lipschitz condition. Without this inclusion guarantee, the robust decrease condition enforced by the LP cannot be guaranteed to hold for all admissible dynamics, directly undermining the certification claim.

    Authors: We agree with the referee that an explicit demonstration of the inclusion property is essential for the certification claim. The current manuscript constructs the polyhedral set using facet inequalities derived from the Lipschitz condition at each data point, but we did not include a formal proof. In the revised manuscript, we will add a detailed proof in Section 3.2 showing that any vector field satisfying the pointwise data and the global Lipschitz bound must satisfy all the facet inequalities of our uncertainty set. This proof will be based on the triangle inequality for the Lipschitz constant and the fact that the facets are chosen as supporting hyperplanes to the possible range at each point. With this addition, the robust decrease condition will be guaranteed to apply to all admissible dynamics. revision: yes

  2. Referee: Section 4: The LP formulation for selecting the PWA Lyapunov function is described at a high level, but it is not shown how the robust decrease condition over the entire polyhedral set is reduced to a finite number of linear constraints (e.g., via vertex or facet enumeration). This step is load-bearing for the claim that the synthesized function certifies a valid RoA.

    Authors: The referee correctly identifies that the reduction to finite constraints is critical. In the manuscript, we mention that the robust condition is enforced over the polyhedral uncertainty set, but the details of how this is achieved via vertex enumeration are not fully elaborated. We will revise Section 4 to include the complete derivation: for each affine region of the PWA Lyapunov function, the decrease condition must hold for all vectors in the polytope. Since the set is polyhedral and compact, the maximum violation occurs at a vertex. Therefore, it suffices to impose the linear decrease inequalities at each vertex of the uncertainty polytope. We will add this explanation along with the corresponding finite set of constraints that are fed into the LP solver. This will make the load-bearing step explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses independent data and Lipschitz bounds to construct verifiable uncertainty set

full rationale

The paper's chain proceeds from externally supplied point-wise vector-field evaluations plus independently known Lipschitz constants to a polyhedral outer approximation of admissible dynamics, followed by an LP that selects a continuous PWA Lyapunov function enforcing a robust decrease condition over that set. None of the load-bearing steps reduces by definition or by self-citation to a quantity fitted to the final certified region; the inputs (data points and Lipschitz bounds) are not derived from the output RoA or from any prior result by the same authors that would close a self-referential loop. The method is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of accurate point-wise vector-field evaluations, valid Lipschitz bounds, and the solvability of the linear program for a suitable PWA function; no new physical entities are introduced.

free parameters (1)
  • State-space partition for PWA function
    The piecewise affine structure requires a user- or algorithm-chosen partition of the state space whose fineness affects both conservatism and computational cost.
axioms (1)
  • domain assumption Point-wise vector field evaluations are exact and the supplied Lipschitz bounds are valid over the considered domain.
    This premise is invoked to construct the polyhedral uncertainty set that contains all admissible dynamics.

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