Koopman Operator via Infinite Input Sequences

• Koopman operator via infinite input sequences is a framework that lifts control systems into infinite-dimensional spaces to combine state and input histories for linear analysis of nonlinear dynamics.
• It constructs observables in a product Hilbert space, enabling scalable, data-driven identification and predictive control through finite-dimensional operator approximations.
• The approach provides explicit error bounds and unifies operator families, making it applicable to areas ranging from oscillator modeling to real-time safety control in complex systems.

The Koopman operator via infinite input sequences is a major extension of the classical Koopman theory, enabling linear representations of nonlinear controlled or open dynamical systems on infinite-dimensional function spaces that encode both state and input histories. This formalism provides the mathematical foundation for advanced system identification, prediction, and control synthesis methods that accommodate persistent (even unbounded) exogenous and feedback-driven input signals.

1. The Koopman Operator and Infinite Input Sequences

The classical Koopman operator is defined for autonomous (input-free) nonlinear dynamical systems by acting on scalar-valued observables (functions on the state space) via composition with the dynamics. In the presence of control inputs or disturbances, the operator must be reformulated, as the state evolution now depends on an entire sequence of inputs. The principal mathematical innovation is to “lift” the control system to a space of functions over $X \times \ell(U)$, where $X$ is the state space and $\ell(U)$ is the space of all infinite input sequences.

For a discrete-time control system $x_{k+1} = T(x_k, u_k)$, the “lifted” system is defined on $(x, \mathbf{u}) \in X \times \ell(U)$ with the evolution map

$$(x, \mathbf{u})^+ = (T(x, u_0), S_{\text{left}}(\mathbf{u}))$$

where $S_{\text{left}}$ is the left-shift operator on infinite input sequences. The Koopman operator $K_\infty$ then composes observables $f: X \times \ell(U) \to \mathbb{R}$ as $K_\infty f = f \circ T_\infty$. This formulation incorporates the entire (potentially infinite) control trajectory into the operator framework, thus enabling analysis and synthesis for arbitrary exogenous or feedback input processes (Haseli et al., 16 Oct 2025).

2. Equivalence with Operator Families and the Koopman Control Family

A fundamental theoretical result is that the Koopman operator defined via infinite input sequences ($K_\infty$) is equivalent, under mild conditions, to the “Koopman control family” (KCF) framework. The KCF comprises a collection of Koopman operators $\{K_{u^*}\}_{u^* \in U}$ indexed by constant control values, each evolving observables defined on the state space. The paper (Haseli et al., 16 Oct 2025) shows that for function spaces with appropriate “extension” and “restriction” mappings, there is an explicit isomorphism between functions on $X$ (the KCF framework) and control-independent subspaces of $X \times \ell(U)$ (the infinite input sequence framework). For any function $f \in \mathcal{F}$ and any sequence $(u_0, u_1, ...)$, it holds that

$$f(x_k) = [K_{u_0} \circ K_{u_1} \circ ... \circ K_{u_{k-1}} f](x_0) = [K_\infty^k(f^\text{ext})](x_0, \mathbf{u})$$

where $f^\text{ext}$ denotes the function extension. This equivalence ensures that either framework yields the same prediction on system trajectories, providing theoretical justification for their interchangeable use (Haseli et al., 16 Oct 2025).

3. Construction of Generalized Koopman Operators in Product Spaces

The generalization of the Koopman operator to systems with inputs relies on constructing observable functions in a product Hilbert space, often realized as $H = H_x \otimes H_u$, the tensor product between the Hilbert spaces of state and input observables. The combined observable takes the form

$\Psi(x, u) = \Psi_x(x) \otimes \Psi_u(u)$

and the generalized Koopman operator $\mathcal{K}$ is a linear map from this product space to the state observable space such that

$$\Psi_x[F(x, u)] = \mathcal{K} \big(\Psi_x(x) \otimes \Psi_u(u)\big)$$

(Lazar, 10 Aug 2025). This bilinear representation enables capturing arbitrary nonlinear, input-dependent dynamics via a linear (infinite-dimensional) operator.

The product Hilbert space construction facilitates scalable data-driven computation of Koopman operators. Lifted data matrices are constructed using the Khatri–Rao (column-wise Kronecker) product, enabling practical computation of finite-dimensional operator approximations, independent sampling of state and input spaces, and tractable regression-based identification strategies (Lazar, 10 Aug 2025).

4. Implications for Data-driven System Identification and Control

Formulating Koopman operators on infinite input sequences forms the basis for several modern identification and control strategies:

• Time-series forecasting and system identification: Observation spaces that include time-delay coordinates or “Hankel” embeddings naturally encode long or infinite input and state sequences, as used extensively in DMD, EDMD, and kernel-based approximations (Bevanda et al., 2021, Mezic, 2020). The use of RKHS-based nonparametric regression (e.g., (Bevanda et al., 12 May 2024)) or deep neural network observables (e.g., (Meng et al., 2023)) allows for learning high-fidelity, finite-rank operator approximations from data, with quantifiable error and convergence properties.
• Predictive control and trajectory synthesis: The infinite sequence viewpoint permits direct computation of future trajectories and output predictions as iterates of the Koopman operator. Data-enabled predictive control methods, such as those built on the “nonlinear fundamental lemma” (generalizing Willems’ fundamental lemma via the bilinear Koopman embedding), allow the use of observed trajectory data to synthesize optimal or constrained predictions for controlled nonlinear systems (Lazar, 10 Aug 2025).
• Closed-loop modeling and feedback representation: By explicitly incorporating feedback laws as arguments to the Koopman-Nemytskii operator, it is possible to obtain linear (infinite-dimensional) models of closed-loop systems—enabling analysis and synthesis tasks that require reasoning about feedback interconnections and the effect of arbitrary input sequences or feedback policies (Tang, 24 Mar 2025).

5. Error Bounds, Approximate Closure, and Convergence

A central challenge in implementing Koopman operator methods via finite approximations is the quantification of error and convergence relative to the true infinite-dimensional operator. Research has established explicit and uniform error bounds for several approximation schemes:

• Bernstein polynomial approximation: Uniform (supremum norm) bounds in terms of the modulus of continuity of the observable are provided for both univariate and multivariate systems, ensuring robust trajectory prediction over long horizons (Yadav et al., 4 Mar 2024).
• Logistic function dictionaries: Approximate closure and global bounds for finite sets of “state-inclusive” observables built from multivariate logistic functions can be ensured by adjusting basis parameters such as steepness and mesh resolution (Johnson et al., 2017).
• RKHS regression and kernel extensions: The density of the RKHS and properties of the kernel-enabled operator regression guarantee that any Koopman operator can be approximated arbitrarily well by finite-rank truncations, with error bounds explicitly related to the kernel bandwidth, data coverage, and regularization parameters (Bevanda et al., 12 May 2024, Zanini et al., 2021).

These results are critical for rigorous application in observer synthesis, robust output feedback, and model validation.

6. Current Challenges and Future Perspectives

Extending the Koopman operator via infinite input sequences raises several open challenges:

• Selection of observable dictionaries: There is no general framework for selecting an optimal set of observable functions that guarantee an invariant subspace under composition with the system dynamics and the infinite input shift operator. This remains a hindrance for achieving globally accurate and robust finite-rank approximations (Bevanda et al., 2021).
• Computational scalability: Infinite-dimensional regression and operator learning are computationally intensive. Techniques such as kernel truncation, sketching (Nyström methods), and deep representation learning mitigate this challenge, but balancing expressive power and tractability is unresolved in high dimensions (Bevanda et al., 12 May 2024, Meng et al., 2023).
• Unified theoretical foundation: The equivalence between Koopman extensions via infinite input sequences and operator families, as rigorously established in (Haseli et al., 16 Oct 2025), provides a basis for the unification of diverse modeling and control approaches. Nevertheless, further research is needed to leverage this equivalence for practical synthesis methods and for the integration of robustness and uncertainty modeling.
• Extensions to stochastic, hybrid, and distributed systems: While the core theory addresses deterministic, time-invariant, continuous, or discrete systems, many applications require generalizations to stochastic, hybrid (with switches or jumps), or networked interconnected dynamics. Current frameworks remain largely confined to deterministic or well-structured classes.

7. Applications and Demonstrated Performance

The Koopman operator via infinite input sequences and its approximations have been validated across a range of applications:

• Nonlinear oscillator and bistable switch modeling: State-inclusive logistic dictionaries have successfully captured the nonlinear phase-space structure of the Van der Pol oscillator and protein toggle switches, with explicit error control (Johnson et al., 2017).
• Hybrid and closed-loop systems: Structured identification with known controller models enables safe and accurate learning in feedback systems (e.g., rotary inverted pendulum, Duffing oscillator) (Dahdah et al., 2023).
• High-dimensional and partial differential systems: Modular product Hilbert space lifting has enabled scalable identification and prediction in controlled PDEs and high-dimensional fluid flows (Lazar, 10 Aug 2025, Mauroy, 2021).
• Real-time safety and control: Deep Koopman operator models, equipped with control barrier functions, have realized real-time safety-assured control in autonomous vehicle platforms, integrating learned prediction, safety constraint encoding, and optimal action selection via quadratic programming (Chen et al., 16 May 2024).

These results demonstrate the practical viability of the infinite input sequence formalism and highlight the central importance of operator-theoretic formulation in contemporary nonlinear system analysis and control.

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This entry consolidates the current state of knowledge on the Koopman operator via infinite input sequences, the equivalence of Koopman control frameworks, their mathematical structure, and their impact on data-driven system analysis and control (Haseli et al., 16 Oct 2025, Lazar, 10 Aug 2025, Tang, 24 Mar 2025, Johnson et al., 2017, Bevanda et al., 2021, Yadav et al., 4 Mar 2024, Bevanda et al., 12 May 2024).