--- title: Koopman Operator Formalism url: https://www.emergentmind.com/topics/koopman-operator-formalism type: topic --- # Koopman Operator Formalism 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 [1907.10807], [2511.06063], [2110.08442]. 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 [1907.10807], [2110.08442], [2103.12458]. ## 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 [1907.10807], [1706.00159], [2110.08442]. ## 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 [1801.07799], [2103.13752], [2511.06063]. 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 [1907.10807], [2407.16542]. ## 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 [2511.06063]. - Invariant sets, attraction basins, and ergodic partitions can be identified via level sets of eigenfunctions with eigenvalue 1 or close to 1 [1907.10807], [1706.00159]. - Data-driven error bounds on the learned Koopman operator lead to probabilistic stability assessments [2511.06063]. ## 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 [2510.15166]. - 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 [2503.18269]. - 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 [2211.07112]. - 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 [2207.12132]. ## 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 [1907.10807]. - **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 [2506.21844], [2101.05873]. - **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 [1608.08734]. - **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 [2306.15601]. Further, the Koopman–van Hove formalism extends operator methods to classical Hamiltonian mechanics, resolving gauge covariance issues [2107.06004]. - **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 [2510.05863]. ## 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 [2110.08442], [2506.21844]. - 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 [2101.05873], [2506.21844]. - 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 [1801.07799], [2407.16542], [2103.13752]. --- **References:** - [1907.10807] On the Koopman operator of algorithms - [1801.07799] Koopman spectra in reproducing kernel Hilbert spaces - [2511.06063] Koopman Operator for Stability Analysis: Theory with a Linear–Radial Product Reproducing Kernel - [2110.08442] Koopman Operator Theory for Nonlinear Dynamic Modeling using Dynamic Mode Decomposition - [2510.05863] Analog and Symbolic Computation through the Koopman Framework - [2101.05873] Data-driven learning for the Mori-Zwanzig formalism: a generalization of the Koopman learning framework - [2207.12132] Koopman Form of Nonlinear Systems with Inputs - [1706.00159] Applied Koopman Operator Theory for Power Systems Technology - [2211.07112] Koopman Bilinearization of Nonlinear Control Systems - [2503.18269] Koopman-Nemytskii Operator: A Linear Representation of Nonlinear Controlled Systems - [1608.08734] An operator-theoretic viewpoint to non-smooth dynamical systems: Koopman analysis of a hybrid pendulum - [2506.21844] Koopman operator-based discussion on partial observation in stochastic systems - [2103.13752] Estimating Koopman operators for nonlinear dynamical systems: a nonparametric approach - [2407.16542] Data-driven Koopman operator predictions of turbulent dynamics in models of shear flows - [2510.15166] Two Roads to Koopman Operator Theory for Control: Infinite Input Sequences and Operator Families - [2306.15601] Hybrid Koopman C*-formalism and the hybrid quantum-classical master equation - [2103.12458] Koopman operator framework for spectral analysis and identification of infinite-dimensional systems - [2107.06004] The dynamics of the Koopman-van Hove angular and linear momentum operators - [2409.18133] The Dirac operator for the pair of Ruelle and Koopman operators, and a generalized Boson formalism Source: https://www.emergentmind.com/topics/koopman-operator-formalism