Koopman Operator Formalism

• Koopman Operator Formalism is a linear operator method that analyzes nonlinear dynamics by studying the evolution of observables rather than the states directly.
• It employs spectral decomposition to express observables as sums of eigenfunctions, revealing key decay rates, oscillatory behaviors, and invariant sets.
• Data-driven techniques like DMD and kernel methods approximate the infinite-dimensional Koopman operator to enable stability analysis, control design, and predictive modeling in complex systems.

The Koopman operator formalism provides a linear, operator-theoretic framework for the analysis of nonlinear dynamical systems by focusing on the evolution of observables—functions of the system state—rather than the states themselves. This approach transforms the nonlinear evolution in finite-dimensional state space to a linear albeit infinite-dimensional evolution in function space. Originating in ergodic theory and spectral analysis, the Koopman formalism has been expanded to encompass data-driven modeling, control, and the study of complex systems across deterministic, stochastic, discrete, and continuous domains (Dietrich et al., 2019, Tang et al., 8 Nov 2025, Snyder et al., 2021). The recent proliferation of research highlights its utility in the analysis of numerical algorithms, stability certification, data-driven modeling, and extensions to systems with inputs and feedback.

1. Mathematical Structure of the Koopman Operator

Given a dynamical system evolving on a state space $X \subset \mathbb{R}^d$:

• Discrete-time dynamics: $x_{n+1} = T(x_n)$, $T:X \to X$
• Continuous-time dynamics: $\dot{x} = v(x)$, generating flow $\varphi^t(x)$

The Koopman operator is defined on a space of observables $g:X \to \mathbb{C}$:

• Discrete-time: $(U g)(x) = g(T(x))$
• Continuous-time: $(K^t g)(x) = g(\varphi^t(x))$

Both $U$ and $K^t$ are linear operators (composition operators) on the infinite-dimensional space of observables (e.g., $L^2(X, \mu)$). The continuous-time operator forms a strongly continuous $C^0$-semigroup, with infinitesimal generator

$$\mathcal{A}g = \lim_{t \to 0} \frac{K^t g - g}{t}, \quad \text{so} \quad \partial_t g(x,t) = \mathcal{A}g(x,t) = \langle \nabla g(x), v(x) \rangle$$

This approach enables the analysis of nonlinear, high-dimensional, or even infinite-dimensional systems by exploiting the linear structure of the Koopman operator (Dietrich et al., 2019, Snyder et al., 2021, Mauroy, 2021).

2. Spectral Theory and Modal Decomposition

The spectral decomposition of the Koopman operator underpins its analytical power:

• Eigenfunctions and eigenvalues:
  • Discrete: $U \phi_k = \lambda_k \phi_k$
  • Continuous: $K^t \phi_k = e^{t \omega_k} \phi_k$
• Any observable $g$ in the span of $\{\phi_k\}$ admits Koopman mode expansion:

$g(x) = \sum_k c_k \phi_k(x)$

• Evolution:

$$U^n g(x) = \sum_k c_k \lambda_k^n \phi_k(x), \qquad K^t g(x) = \sum_k c_k e^{t \omega_k} \phi_k(x)$$

The coefficients $c_k$ are Koopman modes, providing physically interpretable decompositions of observables in terms of invariant sets, decay rates, and oscillatory behaviors (Dietrich et al., 2019, Susuki et al., 2017, Snyder et al., 2021).

3. Data-Driven and Kernel Approximations

Finite-dimensional approximations of otherwise infinite-dimensional Koopman operators are achieved via Dynamic Mode Decomposition (DMD) and its extensions, notably Extended DMD (EDMD). Given snapshot pairs $\{(x_j, y_j = T(x_j))\}$ and a dictionary $\Psi$ of $N_D$ observables:

• Build design matrices $\Psi(X)$ and $\Psi(Y)$.
• Least-squares Galerkin approximation:

$\tilde{K} = \Psi(Y)\, \Psi(X)^\dagger$

• For kernelized/functional settings, a reproducing kernel Hilbert space (RKHS) is used, allowing observables to be elements of the RKHS and allowing flexible, nonparametric approximations that automatically adapt to the geometry of the data (Das et al., 2018, Zanini et al., 2021, Tang et al., 8 Nov 2025).

The choice of dictionary (monomials, RBFs, data-driven bases) critically determines approximation fidelity. In high-dimensional systems with attracting low-dimensional manifolds, data-adapted bases or kernel methods enable the Koopman operator to approximate the reduced dynamics efficiently (Dietrich et al., 2019, Constante-Amores et al., 2024).

4. Stability, Invariant Manifolds, and Operator-Based Certificates

The spectral radius of the Koopman operator encodes key global stability properties:

• If all eigenvalues of the Koopman operator (or its finite-dimensional approximation) are inside the unit disk, the underlying system is asymptotically stable.
• RKHS theory with linear–radial product kernels yields a well-posed Koopman operator that is sensitive to both local equilibrium stability and global smoothness. The location of the spectral radius (relative to the unit disk) provides a stability certificate, and a kernel Lyapunov equation can be formulated in the RKHS (Tang et al., 8 Nov 2025).
• Invariant sets, attraction basins, and ergodic partitions can be identified via level sets of eigenfunctions with eigenvalue 1 or close to 1 (Dietrich et al., 2019, Susuki et al., 2017).
• Data-driven error bounds on the learned Koopman operator lead to probabilistic stability assessments (Tang et al., 8 Nov 2025).

5. Koopman Operator in Control, Feedback, and Input-Driven Systems

Koopman formalism has been extended to handle controlled, open-loop, input-affine, and closed-loop feedback systems:

• For open-loop input systems, two equivalent representations exist: (i) a single Koopman operator acting on input-sequence-augmented state space, and (ii) a family of Koopman operators, each corresponding to a parametrized input value. Both yield the same predictions for control-independent observables under closure assumptions (Haseli et al., 16 Oct 2025).
• The Koopman-Nemytskii operator provides a linear mapping from a product RKHS of states and feedback laws to an RKHS of states, rigorously enabling closed-loop feedback analysis and prediction. This framework offers operator learning algorithms with error bounds for trajectory and cost predictions (Tang, 24 Mar 2025).
• For control-affine systems, the infinitesimal generator of the Koopman operator yields a bilinear structure on an infinite-dimensional Lie group. This leads to a coordinate-free, global framework for controllability analysis, geometric feedback linearization, and Lyapunov theory (Zhang et al., 2022).
• Input-driven systems give rise to linear-parameter-varying (LPV) Koopman models, where the input matrix in the lifted system is state- or input-dependent, yielding precise structure for model identification and gain-scheduling control (Iacob et al., 2022).

6. Applications and Generalizations

The Koopman operator formalism has demonstrated broad utility across domains:

• Numerical Algorithms: Analysis and data-driven acceleration of gradient descent, Nesterov, and Newton–Raphson methods; spectral decomposition reveals contraction rates, basins of attraction, and global convergence structures (Dietrich et al., 2019).
• Stochastic Systems and Partial Observation: The formalism extends to stochastic systems via expectation over transition kernels, and the incorporation of delay coordinates with RKHS/EDMD dictionaries captures memory effects arising from partial observation, as formally linked to the Mori–Zwanzig projection (Ohkubo, 27 Jun 2025, Lin et al., 2021).
• Hybrid and Non-Smooth Systems: For hybrid systems exhibiting discrete resets (e.g., switched or impacted systems), the Koopman operator remains linear and encodes complex basin geometries and invariant sets (Govindarajan et al., 2016).
• Quantum–Classical and Operator–Algebraic Extensions: The hybrid Koopman C*-formalism defines dynamical evolution in hybrid quantum–classical systems as outer automorphisms of a C*-algebra. This unifies quantum and classical dynamics at the operator level and yields quantum–classical master equations (Bouthelier-Madre et al., 2023). Further, the Koopman–van Hove formalism extends operator methods to classical Hamiltonian mechanics, resolving gauge covariance issues (Messadene, 2021).
• Computation Theory: The resolvent of the Koopman operator characterizes reachability and halting in symbolic and analog computation models, establishing links to recursion theory and computability (Caravelli et al., 7 Oct 2025).

7. Limitations, Open Questions, and Future Directions

• The infinite-dimensional nature of Koopman operators necessitates finite-dimensional (data-driven) approximations whose accuracy depends critically on dictionary selection, RKHS properties, and data geometry.
• Identifying Koopman-invariant subspaces remains challenging, especially in systems with continuous or mixed spectra.
• Extensions to stochastic, hybrid, or time-varying systems, as well as systematic observer/estimator design in the lifted space, remain active research areas (Snyder et al., 2021, Ohkubo, 27 Jun 2025).
• The theoretical connection to the Mori–Zwanzig formalism points toward systematic treatment of memory and partial observation; further quantitative analyses of error propagation and projection-induced information loss are ongoing (Lin et al., 2021, Ohkubo, 27 Jun 2025).
• Recent research stresses the importance of kernel-based, deep learning-based, or delay embedding strategies for scalable, robust operator learning in large or partially observed systems (Das et al., 2018, Constante-Amores et al., 2024, Zanini et al., 2021).

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References:

• (Dietrich et al., 2019) On the Koopman operator of algorithms
• (Das et al., 2018) Koopman spectra in reproducing kernel Hilbert spaces
• (Tang et al., 8 Nov 2025) Koopman Operator for Stability Analysis: Theory with a Linear–Radial Product Reproducing Kernel
• (Snyder et al., 2021) Koopman Operator Theory for Nonlinear Dynamic Modeling using Dynamic Mode Decomposition
• (Caravelli et al., 7 Oct 2025) Analog and Symbolic Computation through the Koopman Framework
• (Lin et al., 2021) Data-driven learning for the Mori-Zwanzig formalism: a generalization of the Koopman learning framework
• (Iacob et al., 2022) Koopman Form of Nonlinear Systems with Inputs
• (Susuki et al., 2017) Applied Koopman Operator Theory for Power Systems Technology
• (Zhang et al., 2022) Koopman Bilinearization of Nonlinear Control Systems
• (Tang, 24 Mar 2025) Koopman-Nemytskii Operator: A Linear Representation of Nonlinear Controlled Systems
• (Govindarajan et al., 2016) An operator-theoretic viewpoint to non-smooth dynamical systems: Koopman analysis of a hybrid pendulum
• (Ohkubo, 27 Jun 2025) Koopman operator-based discussion on partial observation in stochastic systems
• (Zanini et al., 2021) Estimating Koopman operators for nonlinear dynamical systems: a nonparametric approach
• (Constante-Amores et al., 2024) Data-driven Koopman operator predictions of turbulent dynamics in models of shear flows
• (Haseli et al., 16 Oct 2025) Two Roads to Koopman Operator Theory for Control: Infinite Input Sequences and Operator Families
• (Bouthelier-Madre et al., 2023) Hybrid Koopman C*-formalism and the hybrid quantum-classical master equation
• (Mauroy, 2021) Koopman operator framework for spectral analysis and identification of infinite-dimensional systems
• (Messadene, 2021) The dynamics of the Koopman-van Hove angular and linear momentum operators
• (Braucks et al., 2024) The Dirac operator for the pair of Ruelle and Koopman operators, and a generalized Boson formalism