⚡ electrical engineering

Koopman-based Estimation of Lyapunov Functions: Theory on a Reproducing Kernel Hilbert Space

This paper establishes a theoretical framework for estimating Lyapunov functions with a specified decay rate for nonlinear systems by leveraging the Koopman operator on a reproducing kernel Hilbert space, where the solution is derived from a unique operator algebraic Lyapunov equation and its statistical error bounds are rigorously established via kernel extended dynamic mode decomposition.

Original authors: Wentao Tang, Xiuzhen Ye

Published 2026-03-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Wentao Tang, Xiuzhen Ye

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Predicting the Future of a Chaotic System

Imagine you are trying to predict the path of a leaf swirling in a turbulent river. The water's movement is nonlinear and chaotic; it twists, turns, and speeds up in ways that are incredibly hard to describe with simple math.

For a long time, engineers have tried to solve this by "linearizing" the system—pretending the river is a straight, calm canal so they can use simple rules to predict where the leaf will go. But this only works for a tiny moment. If you try to predict the leaf's path too far into the future, the simple rules fail, and the prediction goes off the rails.

The Koopman Operator is a clever mathematical trick that says: "What if we stop looking at the leaf, and instead look at the entire river's temperature, color, and speed all at once?" By lifting the problem into a higher dimension (a "function space"), this chaotic, twisting river can be described as a simple, straight line. It turns a messy, non-linear problem into a clean, linear one.

The Problem: The "Ghost" in the Machine

The paper addresses a specific headache with this trick. While the Koopman operator makes the math look linear, it's hard to know if the system is actually stable (will the leaf eventually settle in a calm pool?) or unstable (will it get swept over a waterfall?).

In standard math, if a number is less than 1, things settle down. But with the Koopman operator, the "numbers" (spectrum) depend entirely on how you choose to look at the system. If you choose the wrong "lens" (mathematical space), the operator might look stable even when the system is about to crash, or vice versa. It's like looking at a spinning top through a funhouse mirror; the reflection might look steady, but the top is actually wobbling dangerously.

The Solution: A Special Lens and a "Safety Net"

The authors (Tang and Ye) propose a new way to fix this. They introduce two main innovations:

1. The "Linear-Radial" Lens

They designed a special mathematical "lens" (called a Linear-Radial Product Kernel) to view the system.

The Analogy: Imagine you are looking at a spinning top. A normal lens just sees the blur. This special lens has two parts:
- Linear part: It pays close attention to the very center (the equilibrium point), ensuring it knows exactly where the "stable spot" is.
- Radial part: It pays attention to the smoothness of the air around it, ensuring the math doesn't break down at the edges.
The Result: When they use this specific lens, they proved a golden rule: If the system is stable, the math numbers will always be less than 1. This finally connects the math directly to the physical reality of stability.

2. The "Safety Net" (Lyapunov Function)

Once they know the system is stable, they need to prove how stable it is and draw a map of the safe zone. In engineering, this map is called a Lyapunov Function.

The Analogy: Think of a Lyapunov function as a safety net or a valley. If you drop a ball anywhere in the valley, it will naturally roll down to the bottom (the stable point). The steeper the sides of the valley, the faster it settles.
The Innovation: Usually, finding the exact shape of this valley for a chaotic system is like trying to sculpt a mountain out of fog. The authors created a formula (an Operator Algebraic Lyapunov Equation) that acts like a 3D printer. You feed it the "decay rate" (how fast you want the system to settle), and the formula prints out the exact shape of the safety net.

How They Do It with Data (No Blueprints Needed)

The best part? You don't need to know the physics of the river or the equations of the leaf. You just need data.

Watch and Learn: They watch the system (the river) for a while, recording where things go from point A to point B.
The "kEDMD" Trick: They use a machine learning technique called kEDMD (Kernel Extended Dynamic Mode Decomposition). Think of this as a super-smart pattern recognizer. It looks at the data and builds a "shadow model" of the Koopman operator.
The Safety Check: They plug this shadow model into their "3D printer" formula.
The Guarantee: The paper proves that if you have enough data, the safety net they print will be almost identical to the real one. The more data you feed it, the smaller the error becomes.

The Real-World Test

To prove it works, they tested it on a famous chaotic system called the Brusselator (a model for chemical reactions) and a Liénard equation (a model for oscillating circuits).

They didn't tell the computer the equations.
They just fed it random trajectories.
The computer successfully learned the system, confirmed it was stable, and drew a map (the Lyapunov function) that matched the true physics almost perfectly.

Summary in One Sentence

This paper gives engineers a new, data-driven "magic lens" that turns chaotic, unpredictable systems into simple, linear ones, allowing them to mathematically prove stability and draw a safety map without needing to know the underlying physics equations.

1. Problem Statement

The paper addresses the challenge of performing stability analysis and control for nonlinear systems using data-driven methods. While the Koopman operator offers a powerful framework for "global linearization" of nonlinear dynamics, its application to stability analysis faces two major theoretical hurdles:

Spectrum-Stability Ambiguity: In finite-dimensional linear systems, stability is determined by eigenvalues lying within the unit disk. However, for Koopman operators defined on standard function spaces, the relationship between the operator's spectrum and the system's stability is often elusive or non-informative (e.g., the spectral radius may remain $\ge 1$ even for stable systems depending on the kernel choice).
Lyapunov Function Construction: There is a lack of a rigorous, universal theory to construct Lyapunov functions directly from the Koopman operator, particularly when the operator must be estimated from data.

The authors aim to bridge this gap by establishing a rigorous link between the Koopman spectrum and stability within a Reproducing Kernel Hilbert Space (RKHS) and utilizing this link to estimate Lyapunov functions with guaranteed error bounds.

2. Methodology

The proposed methodology integrates operator theory, kernel methods, and statistical learning theory through the following steps:

A. Function Space Selection: Linear-Radial Product Kernel

To ensure the Koopman operator is well-defined and its spectrum reflects stability, the authors define the operator on a specific RKHS generated by a linear–radial product kernel:
$\kappa(x, x') = (x \cdot x') \cdot \kappa_{=}(x, x')$

$\kappa_{=}$ (Radial): A Wendland radial kernel that ensures the RKHS is equivalent to a Sobolev space ( $W^{s,2}$ ), capturing the smoothness of the dynamics.
Linear Component: The term $(x \cdot x')$ ensures the space contains linear functions near the equilibrium point (the origin).
Result: This construction creates a space where functions are locally linear near the origin and globally regular. Crucially, this allows the authors to prove that if the origin is an asymptotically stable equilibrium, the spectral radius of the Koopman operator $A$ satisfies $r(A) < 1$ .

B. Operator Algebraic Lyapunov Equation (ALE)

The authors formulate the Lyapunov function $v(x)$ and the decay rate function $w(x)$ as kernel quadratic forms defined by positive operators $P$ and $Q$ on the RKHS:
$v(x) = \langle \kappa(x, \cdot), P\kappa(x, \cdot) \rangle$
$w(x) = \langle \kappa(x, \cdot), Q\kappa(x, \cdot) \rangle$
They establish that finding a Lyapunov function with a specific decay rate is equivalent to solving an Operator Algebraic Lyapunov Equation (ALE):
$APA^* - P = -Q$
where $A^*$ is the adjoint of the Koopman operator.

Theoretical Guarantee: The paper proves that if $r(A) < 1$ and $Q$ is a positive Hilbert-Schmidt operator, the ALE has a unique solution $P$ given by a convergent infinite series: $P = \sum_{t=0}^{\infty} A^{*t} Q A^t$ .

C. Data-Driven Estimation (kEDMD and KRR)

Since the true Koopman operator is unknown, the authors propose a data-driven estimation pipeline:

Koopman Operator Estimation: Use Kernel Extended Dynamic Mode Decomposition (kEDMD) to estimate the operator $A$ from trajectory data. This involves solving a least-squares problem in the RKHS.
Decay Rate Estimation: Use Kernel Ridge Regression (KRR) to estimate the operator $Q$ (representing the decay rate function $w(x)$ ) from data.
Lyapunov Estimation: Solve the discrete matrix version of the ALE using the estimated operators $\hat{A}$ and $\hat{Q}$ to obtain the estimated Lyapunov operator $\hat{P}$ .

3. Key Contributions

Spectrum-Stability Theorem on RKHS: The authors rigorously prove that for a system with an asymptotically stable origin, the spectral radius of the Koopman operator defined on a linear-radial product RKHS is strictly less than 1. This resolves the ambiguity of Koopman spectra in stability analysis.
Existence and Uniqueness of Operator ALE Solution: They establish a converse Lyapunov theorem in the RKHS setting, proving that a unique positive Hilbert-Schmidt operator solution exists for the ALE when the spectral radius condition is met.
Error Bound Analysis: The paper derives a rigorous error bound for the estimated Lyapunov function. It demonstrates that the error in the Lyapunov estimate ( $\|\hat{P} - P\|_{HS}$ ) is proportional to the learning errors of the Koopman operator and the decay rate function. Specifically, the error vanishes as the data size increases.
Unified Framework: The work provides a complete, theoretically grounded framework for data-driven stability analysis, moving beyond heuristic applications of Koopman theory.

4. Results

The methodology was validated through a numerical study on a non-polynomial nonlinear system (the Liénard equation):

System: A continuous-time system discretized with a sampling time $\Delta = 0.2$ .
Data: 50 trajectories sampled from a random distribution in $[-2, 2] \times [-2, 2]$ .
Performance:
- Spectrum: The estimated Koopman spectrum was successfully confined within the unit disk, confirming the stability condition.
- Lyapunov Function: The estimated Lyapunov function $\hat{v}(x)$ closely matched the contour shape of the true Lyapunov function (computed via simulation).
- Accuracy: While the estimate was accurate near the origin, deviations were observed far from the origin (e.g., at $x=(1,1)$ ), attributed to the accumulation of decay rates over long horizons. However, the method successfully captured the nonlinearity, distinguishing it from a simple linear approximation (where $v_{lin}(1,1) \approx 19$ vs. true/estimated $\approx 40$ ).

5. Significance

Theoretical Foundation: This paper moves Koopman-based control from an "art of practicality" to a "theoretically rigorous framework." It provides the necessary mathematical conditions (kernel choice, spectral bounds) to guarantee stability analysis.
Data-Driven Stability: It offers a viable path for verifying stability and constructing Lyapunov functions for complex nonlinear systems where first-principles models are unavailable or intractable.
Future Directions: The authors identify the estimation of domains of attraction (especially for systems with multiple attractors or where the origin is not globally stable) as the next critical research step, laying the groundwork for advanced nonlinear control and observer design.

In summary, this work successfully bridges the gap between the abstract theory of Koopman operators on RKHS and practical, data-driven stability analysis, providing both the theoretical guarantees and the algorithmic tools necessary for implementation.