Koopman Operator Theory Overview

• Koopman Operator is a linear, generally infinite-dimensional operator that encodes observable evolution in nonlinear systems, enabling spectral analysis and robust control.
• Key methodologies such as Dynamic Mode Decomposition (DMD), Extended DMD (EDMD), and kernel-based approaches extract its spectral properties for effective finite-dimensional modeling.
• Applications span model reduction, control design, and system identification, offering practical insights for robotics, power systems, and uncertainty quantification.

The Koopman operator is a linear but generally infinite-dimensional operator that encodes the evolution of observables under nonlinear (and sometimes stochastic) dynamical systems. This operator-theoretic framework shifts the description of dynamical evolution from state-space trajectories to the action of a linear operator on a rich function space of scalar observables, thereby enabling the use of spectral theory, linear control methods, and modern data-driven approximation strategies for the analysis, prediction, and control of nonlinear dynamics. There is now a broad ecosystem of mathematical, computational, and application-focused developments surrounding the Koopman operator, with core methodologies including Dynamic Mode Decomposition (DMD), Extended Dynamic Mode Decomposition (EDMD), kernel-based and neural-network-based approximation schemes, operator-theoretic model reduction, and rigorous connections to spectral geometry, control theory, and stochastic processes. The following sections organize the Koopman operator theory by its mathematical formulation, spectral theory, numerical approximation, extensions, control, and applications, with an emphasis on recent advances and foundational principles.

1. Operator-Theoretic Foundation

Let $f: X \to X$ define a discrete-time dynamical system $x_{n+1} = f(x_n)$, where $X$ is a finite- or infinite-dimensional phase space. The Koopman operator $\mathcal{K}$ is defined via

$(\mathcal{K} g)(x) = g(f(x))$

where $g$ is a scalar-valued observable in a suitable function space (Hilbert, Banach, or RKHS). This construction extends to continuous systems $\dot{x} = F(x)$ with the induced (Perron–Frobenius–Koopman) semigroup $(\mathcal{K}^t g)(x) = g(\varphi^t(x))$, where $\varphi^t$ is the flow. The generator satisfies $(\mathcal{L} g)(x) = \nabla g(x) \cdot F(x)$ in the differentiable case (Snyder et al., 2021, Mezic, 2020, Servadio et al., 2021).

In reproducing kernel Hilbert spaces (RKHS) with feature map $\phi(x) = k(\cdot, x)$, $\mathcal{K}$ inherits the Hilbert structure, acting via

$$(\mathcal{K} h)(x) = \langle h, \phi(f(x)) \rangle_{\mathcal{H}}$$

with domain $$D(\mathcal{K}) = \{ h \in \mathcal{H} : h \circ f \in \mathcal{H} \}$$ (Ishikawa et al., 2024).

In the stochastic and Markovian setting, the Koopman operator generalizes to

$(\mathcal{K} g)(x) = \mathbb{E}[g(X_{t+1})|X_t=x]$

which is a linear, bounded operator on suitable $L^2$ (or RKHS) spaces (Črnjarić-Žic et al., 2017, Hou et al., 27 Jan 2025).

2. Spectral Theory and Koopman Decomposition

The central insight is that $\mathcal{K}$, despite being infinite dimensional and defined for a nonlinear system, is linear: this allows the use of spectral theory. Koopman eigenfunctions $\phi_j$ and eigenvalues $\lambda_j$ are defined by

$\mathcal{K} \phi_j = \lambda_j \phi_j$

with spectral expansion of an observable $g$ (when the spectrum is discrete and the basis complete): $g(x) = \sum_{j=1}^\infty v_j \phi_j(x)$ and after $k$ iterations,

$$g(x_k) = (\mathcal{K}^k g)(x_0) = \sum_{j=1}^\infty \lambda_j^k \phi_j(x_0) v_j$$

Each triple $(\lambda_j, \phi_j, v_j)$ defines a Koopman eigenvalue, eigenfunction, and mode (Snyder et al., 2021, Mezic, 2020).

For function spaces that are Banach algebras closed under pointwise multiplication (e.g., $C(X)$, certain RKHS), the eigenfunctions and spectrum of $\mathcal{K}$ exhibit a multiplicative lattice structure: products of eigenfunctions yield new eigenfunctions with eigenvalues multiplying (subject to domain closure) (Bramburger, 31 Jul 2025). In $L^2$-based settings, lattice closure is more delicate.

The Koopman spectrum can contain a discrete part (point spectrum), leading to modal expansion, and a continuous part, associated with mixing and nontrivial memory effects (as in the Mori–Zwanzig formalism) (Mezic, 2020).

For random and stochastic dynamical systems (RDS), the spectrum, eigenfunctions, and their behavior under composition, expectation, and multiplication become distinctly different, with the lattice property generally failing (Črnjarić-Žic et al., 2017, Bramburger, 31 Jul 2025).

3. Numerical Methods: DMD, EDMD, and Kernel Techniques

Given the infinite-dimensionality of $\mathcal{K}$, practical computations use finite-dimensional surrogates extracted from data:

• Dynamic Mode Decomposition (DMD): Uses snapshots $X$ and $X'$ (successive state measurements). One computes the best-fit operator $A = X' X^+$, to approximate the action of $\mathcal{K}$ on the low-dimensional state subspace.
• Extended DMD (EDMD): Augments the snapshot data with a dictionary of observables $\Psi(x) = [\psi_1(x), \dots, \psi_p(x)]^T$, constructing data matrices in the lifted space and fitting a finite-dimensional Koopman matrix $K$ via least squares or Gramian matrices. The eigen-decomposition of $K$ yields approximate Koopman eigenvalues, eigenfunctions, and modes (Snyder et al., 2021, Servadio et al., 2021, Zanini et al., 2021).
• Kernel-based and Nonparametric Methods: EDMD is integrated into the RKHS framework, using empirical Gram and covariance matrices on large data sets and regularization (Tikhonov or sparsity-promoting), enabling scalable estimation and stability even with data-analogous function spaces (Zanini et al., 2021, Hou et al., 27 Jan 2025, Ishikawa et al., 2024).
• JetDMD: Builds canonical, coordinate-free basis subspaces in RKHS (using jets of derivatives at a point) and orthonormalizes them to yield nearly invariant, globally defined approximation spaces. Finite-dimensional Koopman or generator estimates are constructed via pseudo-inverse regression, with sharp error rates (factorial) for analytic kernels and domains (Ishikawa et al., 2024).
• Stochastic Hankel DMD: For stochastic or Markov dynamics, builds empirical Hankel matrices by leveraging Krylov subspaces of time-lagged data, ensuring consistency via ergodic averages and residual-based filtering (Črnjarić-Žic et al., 2017).

4. Extensions: Infinite-Dimensional, Control Systems, and Geometry

Koopman theory extends seamlessly to:

• Infinite-Dimensional Systems: Nonlinear PDEs represented as semiflows on function spaces admit a Koopman semigroup acting on bounded continuous functionals, with Lie generators accessible through Gâteaux differentiation. Generalized EDMD yields structural identification and model reduction for nonlinear PDEs (Mauroy, 2021).
• Control-Affine and Input-Output Systems: The action of $\mathcal{K}$ is extended to systems with explicit inputs (lifting via control-enriched dictionaries, or via the control-affine generator decomposition). Bilinearization of the Koopman evolution on infinite-dimensional Lie groups provides a global (algebraic) approach to controllability analysis, extending the Lie Algebra Rank Condition (LARC) to functional spaces and enabling feedback linearization on the controllable submanifold without the strict invertibility classically required (Zhang et al., 2022).
• Rigged Hilbert and Spectral Theory: By embedding RKHS subspaces into a Gelfand triple (dense in a dual space), one defines the extended Koopman operator, including a Jordan-Chevalley decomposition, which clarifies the role and extraction of generalized eigenfunctions and the structure of spectral expansions in non-self-adjoint or non-diagonalizable settings (Ishikawa et al., 2024).

5. Model Reduction, Spectral Geometry, and System Identification

Spectral analysis of Koopman operators yields rigorous model reduction and geometric insights:

• Geometry of Invariant Sets: Level sets and joint fibers of eigenfunctions define invariant foliations of the phase space (e.g., isochrons, stable/unstable manifolds), which underpin global linearizations and reveal the structure of the state space (Mezic, 2020, Katayama et al., 2024).
• Exact Finite-Dimensionality: There exist systems (notably those with polynomial, symmetric, or low-order nonlinearities) where a finite set of observables spans an exactly invariant Koopman subspace, permitting finite-dimensional linear representations and linear control design beyond local linearization (Snyder et al., 2021, Mezic, 2020).
• Nonlinear and Kernel Representation Learning: Neural networks, SINDy-type models, and kernel-based dictionaries can be trained to discover (non)linear invariant subspaces, coordinate encodings, or optimal representations for given data, leading to joint state encoding and spectral/Koopman approximation (Mezic, 2020, Sugishita et al., 2024).
• Stability and Bifurcation Theory: By selecting product kernels (linear–radial) that are invariant under composition with dynamics, one ensures the Koopman operator is well defined and bounded. The spectrum of the Koopman operator (in the RKHS) serves as a certificate for asymptotic stability (spectrum inside the unit disk), and bifurcation is accompanied by spectrum crossing the boundary. Statistical error bounds exist for finite-sample kernel EDMD approximations (Tang et al., 8 Nov 2025).

6. Control Design and Uncertainty Quantification via Koopman Operators

Koopman operator approximations enable new advances in nonlinear control:

• Koopman-Linearized Control: By lifting state (and input) variables to an approximately invariant observable space, nonlinear systems are recast as finite-dimensional linear systems, permitting global application of linear quadratic regulation (LQR) and observer synthesis (e.g., Luenberger, Kalman) far beyond equilibrium neighborhoods (Snyder et al., 2021, Dahdah et al., 2024).
• Uncertainty and Robustness: Koopman-based models permit explicit quantification of model uncertainty in the frequency domain (by comparing transfer functions across a population of systems) and the synthesis of robust observers and controllers via mixed $\mathcal{H}_2 / \mathcal{H}_\infty$ optimization, with successful validation on real-world systems (e.g., industrial motor drives) (Dahdah et al., 2024).
• Feedback Linearization and Controllability: Koopman feedback linearization can be applied on controllable submanifolds, bypassing classical requirements and leveraging algebraic/geometric techniques from differential geometry and infinite-dimensional Lie theory (Zhang et al., 2022).
• Multi-Agent and Game-Theoretic Control: Linearizing the dynamics via agent-wise observable lifts enables centralized or Nash-equilibrium synthesis for multi-agent systems with time scale separation, capturing the divergence between social optima and decentralized rational strategies by explicit cross-coupling in the lifted space (Bakker, 18 Jun 2025).

7. Applications, Model Reconstruction, and Algorithmic Ecosystem

The Koopman operator framework has been deployed in a variety of domains:

• Canonical Nonlinear Oscillators: For benchmarking, the Koopman approach captures the spectral structure and geometric foliation in relaxation oscillators and singularly perturbed regimes (e.g., van der Pol, Duffing), as well as in limit cycle and chaotic attractors (Servadio et al., 2021, Katayama et al., 2024, Ishikawa et al., 2024).
• Power Systems: Modal analysis, swing instability precursors, and coherency identification in large power grids are enabled by spectral Koopman decompositions (often via DMD/EDMD) (Susuki et al., 2017).
• Robot Learning and Control: Koopman operators serve as surrogates for nonlinear robot models, enabling real-time control, active learning, and system adaptation across hardware platforms, with systematic approaches to data collection, dictionary selection, and robust MPC formulation (Shi et al., 2024).
• Neural Networks and Model Compression: Nonlinear blocks in deep neural networks can be (partially) replaced by Koopman-linear surrogates with minimal loss in performance and significant compression, using EDMD and tensor-train approximations (Sugishita et al., 2024).
• Stochastic and Infinite-Dimensional Systems: Koopman-based spectral decomposition supports model reduction and system identification in random dynamical systems and nonlinear partial differential equations, with guaranteed convergence theorems and quantified residuals (Črnjarić-Žic et al., 2017, Mauroy, 2021).
• Spectral Estimation in RKHS and Rigged Spaces: Kernel-based and jet-based approaches offer higher sensitivity and precision in the detection of weak or embedded Koopman eigenfrequencies compared to traditional harmonic analysis (DFT/FFT), providing strong identification in mixed-spectrum and noisy systems (Das et al., 2018, Ishikawa et al., 2024).

Koopman operator theory continues to expand into areas of system identification, representation learning, and robust model-based control. Open challenges include systematic encoding of constraints into observable liftings, quantifying representation expressiveness, extension to hybrid and stochastic systems, and the rigorous theory of kernel- and neural-network-induced function spaces.

---

References

• (Snyder et al., 2021) "Koopman Operator Theory for Nonlinear Dynamic Modeling using Dynamic Mode Decomposition"
• (Mezic, 2020) "Koopman Operator, Geometry, and Learning"
• (Servadio et al., 2021) "A Koopman Operator Tutorial with Orthogonal Polynomials"
• (Mauroy, 2021) "Koopman operator framework for spectral analysis and identification of infinite-dimensional systems"
• (Zanini et al., 2021) "Estimating Koopman operators for nonlinear dynamical systems: a nonparametric approach"
• (Zhang et al., 2022) "Koopman Bilinearization of Nonlinear Control Systems"
• (Katayama et al., 2024) "Koopman Analysis of the Singularly-Perturbed van der Pol Oscillator"
• (Tang et al., 8 Nov 2025) "Koopman Operator for Stability Analysis: Theory with a Linear--Radial Product Reproducing Kernel"
• (Ishikawa et al., 2024) "Koopman operators with intrinsic observables in rigged reproducing kernel Hilbert spaces"
• (Sugishita et al., 2024) "Extraction of nonlinearity in neural networks with Koopman operator"
• (Dahdah et al., 2024) "Uncertainty Modelling and Robust Observer Synthesis using the Koopman Operator"
• (Dahdah et al., 2023) "Closed-Loop Koopman Operator Approximation"
• (Susuki et al., 2017) "Applied Koopman Operator Theory for Power Systems Technology"
• (Das et al., 2018) "Koopman spectra in reproducing kernel Hilbert spaces"
• (Črnjarić-Žic et al., 2017) "Koopman Operator Spectrum for Random Dynamical Systems"
• (Bakker, 18 Jun 2025) "Multi-Agent, Multi-Scale Systems with the Koopman Operator"
• (Hou et al., 27 Jan 2025) "Nonparametric Sparse Online Learning of the Koopman Operator"
• (Bramburger, 31 Jul 2025) "On the lattice property of the Koopman operator spectrum"
• (Shi et al., 2024) "Koopman Operators in Robot Learning"
• (Dietrich et al., 2019) "On the Koopman operator of algorithms"