Data-to-Certificates (D2C): Koopman Supereigenfunctions for Stability, Safety, and Control

arXiv:2610.00178 · math.OC, cs.SY, eess.SY · Submitted 2026-09-16 · Read on arXiv

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Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: I'm Rosa, and with me are Dev and Taro, guest researcher.

Dev: Today's paper: "Data-to-Certificates (D2C): Koopman Supereigenfunctions for Stability, Safety, and Control".

Rosa: Traditional dynamical system models, including Koopman operator representations, are fundamentally equality-based, whereas many analysis and control tools rely on inequalities.

Dev: First, who's behind it and why it matters.

Paper summary: Dev: So, to wrap up the discussion on this paper, "Data-to-Certificates (D2C): Koopman Supereigenfunctions for Stability, Safety, and Control," the authors are essentially proposing a way to move away from purely equality-based models in dynamical systems toward an inequality-based framework that learns certificates directly from data. They introduced supereigenfunctions of the Koopman operator as this new generalization of eigenfunctions, which they use to define exponential growth envelopes that bound system behavior.

Rosa: That’s right, and the implication is that we can bypass the need for explicit model construction by learning these certificates from real-world data. This allows us to derive tools for stability analysis and control synthesis that are naturally compatible with inequality-based methods, which is a significant departure from traditional approaches.

Taro: I think the main point is the task-driven representation paradigm; instead of choosing coordinates based on abstract spectral properties, we can pick observables aligned with our specific objective, whether it's stabilization or safety. That flexibility in choosing the certificate is what makes this framework adaptable to different autonomous tasks.

Dev: From my side, it means we have a mathematical pathway to synthesize controllers that actively shape system growth rates toward desired values through inequality shaping, which is something we’ve been striving for in control engineering but often struggle with when the system is nonlinear.

Rosa: And safety-wise, they provide a way to define safe sets that are aware of the actual dynamics within those regions using these risk probes and their corresponding supereigenfunctions. It’s about getting certificates that actually reflect what's happening during operation, not just theoretical possibilities.

Taro: Overall, the impact seems to be providing a flexible, data-driven mechanism for generating formal guarantees—stability proofs or safe constraints—without requiring us to first perfectly model the system's underlying mathematics.

Dev: While the paper shows strong theoretical foundations for this data-to-certificates (D2C) paradigm, one limitation they point out is that in finite horizon applications, there's an explicit residual error epsilon D2C that needs to be quantified. That means as we move toward practical deployment, we still need a solid way to manage the discrepancy between our learned certificate and the true system behavior.

Rosa: Exactly, and for future work, I think we need to focus heavily on verifying how robust these data-driven certificates are when applied outside of a perfectly controlled environment, which is where field robotics becomes critical.

Taro: And I agree; exploring the system's response when it misbehaves under these learned certificate conditions will be key to showing its viability in complex autonomous operations.

Conclusion: Rosa: So, this paper by the authors, "Data-to-Certificates (D2C): Koopman Supereigenfunctions for Stability, Safety, and Control," is about moving away from building rigid mathematical models to learning safety guarantees directly from data using these new supereigenfunctions.

Dev: I see what you mean; it’s about bypassing that whole explicit model construction phase by letting the data define the bounds of system behavior through these inequalities.

Taro: What struck me most is how they use those supereigenfunctions as a way to encode exponential growth envelopes, which gives us certificates for stability and safety right out of the data analysis.

Rosa: That's what I find fascinating because it suggests we might be able to get guarantees for complex robotic systems without having to perfectly map every single variable in a high-dimensional state space.

Dev: From my side, it’s interesting how they connect this directly to comparison dynamics; if those certificates satisfy certain inequalities, we can use them to derive simple linear systems for stability analysis.

Taro: And for autonomy research, the idea of using risk probes to define "dynamics-aware safe sets" based on these resolvent supereigenfunctions is exactly what we need when things go wrong in the field.

Rosa: So, putting it simply, this work offers a new way to generate formal safety and control proofs just by looking at operational data rather than relying solely on theoretical equations.

Dev: It’s quite a leap from classical methods because it shifts the focus from exact equality to bounding behavior using inequalities derived from the Koopman operator's structure.

Taro: The potential impact here is huge because it moves formal verification tools closer to real-world deployment scenarios where the system dynamics are often too complex for traditional analysis.

Rosa: It makes me wonder if this works well outside of a controlled lab setting and for how long we can actually trust these data-derived envelopes in unpredictable environments.

math.OC, cs.SY, eess.SY

Submitted: 2026-09-16

Updated: 2026-09-16

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 84/100

The gist: Traditional dynamical system models, including Koopman operator representations, are fundamentally equality-based, whereas many analysis and control tools rely on inequalities.

Key concepts

Supereigenfunctions: These are functions derived from the Koopman operator that satisfy an inequality (Kfφϕ) ≤ λφϕ instead of an equality. They act like Lyapunov functions by defining exponential growth envelopes that bound system behavior, encoding stability and safety information through these inequalities.
Directional Supereigenfunctions: These are specific supereigenfunctions derived using the multiplicative ergodic theorem (MET). They offer a geometric interpretation based on tangent dynamics, allowing researchers to associate associated rates with Lyapunov exponents, which measure the system's exponential growth or decay along trajectories.
Resolvent-based Construction: This method generates families of supereigenfunctions directly from data using the resolvent of the Koopman generator and a probe function. It allows for computation without needing an explicit system model, making it a scalable, data-driven alternative to traditional PDE or sum-of-squares approaches.
Certificate Coordinate Map: This map transforms the system state into a vector of supereigenfunctions (Φ(x)). By analyzing the componentwise inequality KfΦ(x) ⪯ ΛΦ(x), researchers can compare the system's evolution to a simple linear comparison system, enabling stability analysis and control design.

Terminology

Generalization — 6× in this paper · explained in 13 episodes
Lyapunov function — 6× in this paper · explained in 2 episodes
Verification — 2× in this paper · explained in 2 episodes
World Model — 2× in this paper · explained in 7 episodes
Alignment — 1× in this paper · explained in 1 episode
Closed-Loop Control — 1× in this paper · explained in 3 episodes
Core — 1× in this paper · explained in 2 episodes
Digital Twin — 1× in this paper · explained in 5 episodes
Fidelity — 1× in this paper · explained in 2 episodes
Formal Verification — 1× in this paper · explained in 2 episodes
Robustness — 1× in this paper · explained in 5 episodes
Uncertainty Quantification — 1× in this paper · explained in 8 episodes

Summary

Traditional dynamical system models, including Koopman operator representations, are fundamentally equality-based, whereas many analysis and control tools rely on inequalities. This paper proposes a data-to-certificates (D2C) paradigm that bypasses explicit model construction and directly learns certificates from data by introducing supereigenfunctions of the Koopman operator as an inequality-based generalization of eigenfunctions that define exponential growth envelopes encoding stability, safety, and uncertainty propagation.

Supereigenfunctions as Certificates

The core concept is the introduction of supereigenfunctions, defined by relaxing the spectral equality to a one-sided inequality: (Kfφϕ) ≤ λφϕ. This relaxation fundamentally changes the role of the representation; instead of encoding exact trajectory evolution, supereigenfunctions define exponential growth envelopes that bound system behavior. These objects are operator-theoretic analogues of Lyapunov and value functions, providing certificates for stability and safety. Specifically, if a function satisfies (Kfφϕ) ≤ λφϕ, it implies that along the trajectories of the system st(x), d/dt φ(x(t)) ≤ λφ(x(t)), which leads to the exponential envelope: φ(st(x)) ≤ e λt φ(x). This allows for certificates that are naturally compatible with Lyapunov functions and Hamilton–Jacobi value functions.

Complementary Constructions for Data-Driven Computation

The paper develops two complementary constructions to realize this framework. First, it introduces directional supereigenfunctions, which arise from the multiplicative ergodic theorem (MET), providing a geometric interpretation based on tangent dynamics and associated rates corresponding to Lyapunov exponents. Second, it develops a resolvent-based construction that generates families of supereigenfunctions from user-defined probe functions. This latter approach admits trajectory-based representations and enables computation directly from data, without explicit knowledge of the system, offering a scalable alternative to PDE-based or sum-of-squares approaches.

Certificate Generation via Positive Resolvents

The data-to-certificate (D2C) construction utilizes the resolvent of the Koopman generator. Given a positive Koopman C0-semigroup and a nonnegative probe function g, the resulting observable is defined as φλ = R(λ; Kf)g = Z ∞ 0 e(-λtUtg dt. Theorem 4 establishes that this yields an exact positive certificate satisfying (Kfφλ) = λφλ − g ≤ λφλ, meaning it is a Koopman supereigenfunction with rate λ. For finite-horizon applications, the construction yields approximate certificates where the residual error is explicitly quantified as εD2C, allowing for data-driven and dynamically consistent alternative[s] to classical certificate design.

Application in Stability and Control Synthesis

The resulting framework allows for stability analysis and control synthesis through comparison dynamics. By defining a certificate coordinate map Φ(x) = [φ1(x),..., φm(x)]⊤, the componentwise inequality KfΦ(x) ⪯ ΛΦ(x) (where Λ = diag(λ1,..., λm)) implies that the evolution of these coordinates is dominated by a positive linear comparison system. This mechanism is used to derive:

Stability from contracting certificates: If the certificates satisfy Kfφi ≤ λiφi with λi < 0 and there exist class-K∞ functions α1, α2 such that α1(∥x∥) ≤ 1⊤Φ(x) ≤ α2(∥x∥), then the origin is globally asymptotically stable.
Stabilizing control via inequality shaping: For a control-affine system x˙ = f(x) + G(x)u, stabilization is achieved by enforcing the constraint (KGφi)(x)u ≤ −(λi − βi)φi(x), where desired rates βi < 0. This leads to a convex quadratic program (QP) for control synthesis, ensuring that the controller shapes the growth rates from open-loop values λi to desired negative values βi.

Safety Characterization and Safe Control

The framework extends to safety by using resolvent supereigenfunctions with risk probes r(x) that quantify constraint violations. The infinite-horizon certificate is defined as φλ(x) = Z ∞ 0 e(-λt r(st(x)) dt, interpreted as a discounted risk-to-go. A key result shows that the sublevel set Sλ,T (c) = [x ∈ omega: φλ,T (x) ≤ c] defines a dynamics-aware safe set.

Improvements for AI systems

As a fastidious researcher, I have analyzed this groundbreaking work on Data-to-Certificates (D2C): Koopman Supereigenfunctions for Stability, Safety, and Control. The core innovation is shifting from equality-based Koopman models to inequality-based certificates derived directly from data.

Here are the specific improvements we can implement in AI systems:

The proposed framework allows us to construct robust guarantees directly from trajectory data without requiring an explicit, high-fidelity predictive model of the system dynamics.

To a state estimation or control system that is currently reliant on traditional Lyapunov functions or barrier functions, we can replace them with a set of Supereigenfunctions derived via the Resolvent construction (Theorem 4).
This allows us to define an inequality:

Kf(Φ(x)) ≤ ΛΦ(x)

where is a vector of these certificates and is a diagonal matrix of data-derived rates.

This improved AI system can achieve the following specific capabilities:

To guarantee long-term stability for complex, nonlinear systems (e.g., robotics, autonomous vehicles) by ensuring that the certificate coordinates remain bounded under closed-loop dynamics, even when the underlying model is only learned or approximated via data rollouts.
To design safe control laws that explicitly enforce safety constraints (e.g., obstacle avoidance) by shaping the growth rates of risk-probes (like those derived from Theorem 10). The AI controller will actively modify its input to ensure that a chosen risk sublevel set remains forward invariant, thus guaranteeing that the system never enters dangerous regions defined by the learned safety certificates.
To perform Contraction-Based Synchronization in multi-agent systems (e.g., leader-follower robotics). By using Gramian or MET-based directional supereigenfunctions, the AI can synthesize control laws that enforce the exponential decay of the error between agents' trajectories, leading to high precision synchronization without needing a perfect global model of the system’s coupling dynamics.
To handle uncertainty propagation robustly. Instead of relying on conservative bounding techniques, we can use Approximate D2C certificates (Corollary 1) to quantify exactly how initial state uncertainties grow along the learned trajectory envelope, providing precise, dynamically consistent risk bounds for planning algorithms.
To create highly scalable control synthesis via Convex Quadratic Programming (QP). The AI system will solve a QP at every time step that simultaneously satisfies multiple safety and stability certificates (Theorem 10), allowing it to optimize control inputs based on a vector Lyapunov function constructed from the learned data-driven certificates.

Sources

Discovering the Kalman-Bucy-Koopman Filter