Koopman Operators: Linearizing Nonlinear Dynamics

• Koopman operators are a linear operator framework that represents nonlinear dynamical systems as linear evolutions on infinite-dimensional observable spaces.
• They enable spectral decomposition and data-driven approximations, using methods like DMD and EDMD to extract global dynamical features.
• Applications span control, robotics, and machine learning, offering robust tools for stability certification, feedback linearization, and model reduction.

The Koopman operator provides an operator-theoretic framework that represents the evolution of nonlinear dynamical systems as linear dynamics on spaces of observables. While the underlying state-space evolution may be nonlinear (or even high-dimensional), the induced operator is always linear, enabling spectral analysis, model reduction, stability certification, and data-driven control synthesis. This perspective has advanced both theory and application across control, robotics, machine learning, and high-dimensional modeling.

1. Operator-Theoretic Foundation

Given a dynamical system—continuous-time ($\dot{x} = F(x)$) or discrete-time ($x_{k+1} = T(x_k)$)—the Koopman operator $\mathcal{K}$ acts on a space of observables $g:X\to\mathbb{C}$ by $\mathcal{K}g(x) = g(T(x))$ (discrete) or $\mathcal{K}_t g(x) = g(\Phi^t(x))$ (continuous). This operator is linear even if $T$ or $F$ is nonlinear, and typically infinite-dimensional because the function space of observables is infinite-dimensional (Mezic, 2020, Servadio et al., 2021).

The infinitesimal generator in continuous-time,

$$L_G f = \lim_{t \to 0} \frac{K^t f - f}{t} = G\cdot\nabla f,$$

encodes the instantaneous observable dynamics (Servadio et al., 2021, Das, 2023, Valva et al., 2024).

Koopman theory relies critically on the spectral decomposition of $\mathcal{K}$, which allows one to associate eigenvalues with global dynamical features (fixed points, periodic orbits, stability, mixing), and eigenfunctions with invariant sets, isochrons, and model decompositions (Susuki et al., 2017, Mezic, 2020).

2. Spectral Properties and Lattice Structure

The spectral content of the Koopman operator is central for dynamical analysis. Koopman eigenfunctions $\phi$ satisfy $\mathcal{K} \phi = \lambda \phi$ and generically encode coherent structures. The spectrum $\sigma(\mathcal{K})$ may consist of point, continuous, and residual parts. In measure-preserving or ergodic systems, $\mathcal{K}$ is a unitary operator with purely point and/or continuous spectra (Colbrook et al., 2021, Valva et al., 2023, Bramburger, 31 Jul 2025).

A rigorous lattice property holds for deterministic discrete-time Koopman operators: the spectrum is multiplicatively closed, i.e., if $\lambda, \eta \in \sigma(\mathcal{K})$, then $\lambda\eta \in \sigma(\mathcal{K})$; products of eigenfunctions correspond to products of eigenvalues. However, in stochastic cases (Koopman operator of a Markov kernel), the spectrum is not generally multiplicatively closed, which fundamentally influences the interpretation of spectral objects in data-driven settings (Bramburger, 31 Jul 2025).

For high-dimensional or hybrid systems (e.g., limit cycles with discrete transitions), spectral analysis requires precise construction of observable spaces, often involving Banach or reproducing kernel Hilbert spaces (RKHS) (Katayama et al., 2024, Tang et al., 8 Nov 2025, Ishikawa et al., 2024). Hybrid limit-cycle systems admit globally unique principal eigenfunctions that encode phase-amplitude coordinates and facilitate global analysis (Katayama et al., 2024).

3. Data-Driven Approximation

Finite-data estimation of Koopman operators leverages Dynamic Mode Decomposition (DMD), Extended DMD (EDMD), and their kernel/RKHS extensions. EDMD projects the infinite-dimensional Koopman operator onto a finite dictionary of observables, yielding a finite matrix that can be analyzed via standard spectral techniques. Kernel-based variants replace explicit dictionaries with data-adaptive kernels, producing nonparametric estimators with rigorous convergence guarantees (Zanini et al., 2021, Mezic, 2020, Servadio et al., 2021, Ishikawa et al., 2024, Sznaier, 2021).

For infinite-dimensional systems (e.g., PDEs), one selects continuous functionals as observables and computes finite-dimensional approximations of the semigroup and its generator. Generalized EDMD on bases of functionals $\zeta_j$ yields convergence in the operator norm and recovers, for suitable choices, the underlying nonlinear generator coefficients (Mauroy, 2021).

Residual Dynamic Mode Decomposition (ResDMD) advances spectral computation by explicitly verifying residuals, enabling pseudospectral analysis, preventing spectral pollution, and offering robust convergence rates—even for systems with continuous spectra and high-dimensional data (Colbrook et al., 2021).

Recent work on convex optimization approaches solves for both the Koopman operator and the appropriate finite-dimensional embedding via rank-constrained semi-definite programs, ensuring global optimality (up to relaxation) and scalability via chordal sparsity decomposition (Sznaier, 2021).

4. Reproducing Kernel Hilbert Spaces and Intrinsic Observables

RKHS frameworks supply precise regularity and approximation guarantees. By selecting product kernels (e.g., linear × Wendland radial), one constructs RKHSs that are invariant under the Koopman action, locally encode stability around equilibria, and globally encode regularity (Tang et al., 8 Nov 2025).

Jets (intrinsic observables built from derivatives at fixed points) provide coordinate-invariant local bases for spectral approximation and enable robust Galerkin schemes (JetDMD) (Ishikawa et al., 2024). Rigged Hilbert space approaches generalize the domain of the Koopman operator, enabling spectral analysis on larger spaces (Gelfand triples) and yielding explicit reconstruction and convergence theorems in both discrete and continuous time.

Physics-informed spectral schemes approx the Koopman generator (infinitesimal) via compact, skew-adjoint operators in RKHS, integrating physical vector field information via automatic differentiation. Variational Galerkin eigenproblems solve for unitary-invariant basis features, with strong convergence results in the large-data limit (Valva et al., 2024, Valva et al., 2023).

5. Control, Stabilization, and Feedback Linearization

Koopman-based control schemes leverage the global linearity of the operator to transform nonlinear control problems. In model predictive control (MPC), data-driven lifted linear predictors allow tube-based MPC with robust invariance, recursive feasibility, and pointwise nominal convergence—without requiring that the finite-dimensional Koopman model be an exact surrogate of the full system (Zhang et al., 2021, Snyder et al., 2021, Shi et al., 2024).

The spectrum-stability correspondence is direct: if the Koopman spectrum is confined to the unit circle (discrete-time) or left half-plane (continuous-time), the underlying equilibrium is asymptotically stable; bifurcation corresponds to spectral escape (Tang et al., 8 Nov 2025). Kernel Lyapunov equations constructed in RKHS provide certificates of stability and enable synthesis of feedback laws by solving Riccati-like or Hamilton–Jacobi–Bellman equations in operator coordinates.

Koopman bilinearization generalizes control-affine systems into infinite-dimensional bilinear systems, facilitating operator-theoretic controllability via the de Rham differential and yielding feedback linearization procedures that only require controllability on the reachable submanifold, with local linearization in observable coordinates (Zhang et al., 2022).

6. Extended Representations, Machine Learning, and Applications

Koopman operator theory encompasses finite-dimensional, nonlinear, and learning-based representations. Finite linear representations are possible exactly if a suitable invariant subspace of observables is found (Mezic, 2020). Nonlinear finite-dimensional maps ("eigenmaps," Editor's term) generalize this—learnable via neural networks—enabling construction of surrogates that preserve the linear prediction property in embedded coordinates.

Machine learning approaches (neural autoencoders, temporal Koopman networks, prototype loss architectures) enforce approximate linearity across temporal domains and allow explicit derivation of generalization bounds in terms of KL divergence between conditional distributions in Koopman-embedded space (Zeng et al., 2024). Supervised, unsupervised, and reinforcement learning can exploit Koopman-linear lifts for fast training and robust extrapolation.

Applications span power systems (modal decomposition, stability diagnosis), robot learning (real-time estimation, planning, control), high-dimensional molecular simulation, and the study of numerical algorithms (gradient descent, Nesterov acceleration, Newton–Raphson), with Koopman spectra revealing ergodic partitions, convergence rates, and basin geometry (Susuki et al., 2017, Shi et al., 2024, Dietrich et al., 2019).

7. Open Challenges and Future Directions

Current limitations in Koopman theory include optimal selection of dictionaries and kernels for data-driven estimation, rigorous treatment of stochastic and nonautonomous dynamics (lattice property failure), uncertainty quantification in lifted spaces, and computational cost for high-dimensional or long-memory systems. Spectral pollution, continuous spectrum recovery, and algorithmic stability remain active research areas (Colbrook et al., 2021, Bramburger, 31 Jul 2025, Valva et al., 2023).

Ongoing work explores:

• PDE and high-dimensional system identification via generator approximation and pointwise functionals.
• Nonparametric, kernel-based approaches for large-scale and noisy data settings.
• Rigorous operator norm and spectral measure convergence.
• Feedback control and stabilization in Koopman coordinates.
• Deep learning for embedding and dictionary discovery.
• Quantification and interpretation of continuous spectra, mixing, and bifurcation phenomena.
• Large-scale real-time learning and control in robotics and physical simulation.

These efforts collectively establish Koopman operator theory as an essential framework for global linearization, spectral analysis, stabilization, and control of nonlinear dynamical systems across scientific and engineering disciplines.