Koopman–Nemytskii Operator in Nonlinear Control

• Koopman–Nemytskii operator is a bounded linear operator that embeds nonlinear closed-loop dynamics into reproducing kernel Hilbert spaces using state-feedback laws.
• It generalizes the classical Koopman operator by explicitly incorporating nonlinear feedback, enabling linearization techniques like kernel EDMD for predictive modeling.
• The framework provides rigorous error bounds and supports data-driven control policy optimization and trajectory prediction in complex nonlinear systems.

The Koopman–Nemytskii operator is a bounded linear operator that generalizes the classical Koopman operator to nonlinear controlled systems with feedback. Unlike the standard Koopman operator, which provides a linear but infinite-dimensional representation for autonomous dynamical systems, the Koopman–Nemytskii operator explicitly incorporates nonlinear state-feedback laws, enabling linearization of closed-loop dynamics in reproducing kernel Hilbert spaces (RKHSs). This construction facilitates data-driven predictive modeling and controller synthesis for a broad class of smooth nonlinear control systems via operator learning methodologies such as kernel extended dynamic mode decomposition (kernel EDMD), with rigorous error guarantees (Tang, 24 Mar 2025).

1. Mathematical Formulation and Operator Definition

Let $X\subset \mathbb{R}^{n_x}$ represent the compact state domain and $U$ a compact family of feedback laws $u:X\to\mathbb{R}^{d_u}$. Take $\kappa$ as a Mercer (Wendland) kernel on $X$, yielding RKHS $\mathcal{N}_\kappa(X)\simeq H^s(X)$ for Sobolev index $s$, and $\varpi$ as a universal kernel on $U$ with $\mathcal{N}_\varpi(U)$. The canonical feature maps are

• $U$0, $U$1,
• $U$2, $U$3.

Define $U$4 as the closed-loop map. The Koopman–Nemytskii operator

$U$5

is the unique bounded linear map characterized by

$U$6

Alternatively, equating $U$7 with $U$8 on $U$9, one writes

$u:X\to\mathbb{R}^{d_u}$0

This operator provides a universal, linear embedding of the family of nonlinear closed-loop maps parameterized by the feedback law $u:X\to\mathbb{R}^{d_u}$1.

2. Connections to Koopman, Nemytskii, and Operator Lifting

The Koopman–Nemytskii operator unifies several classical nonlinear operator concepts:

• Autonomous Koopman operator: $u:X\to\mathbb{R}^{d_u}$2, $u:X\to\mathbb{R}^{d_u}$3, describing autonomous dynamics.
• Nemytskii (substitution) operator: $u:X\to\mathbb{R}^{d_u}$4, $u:X\to\mathbb{R}^{d_u}$5, acting on maps from state to action.
• Koopmanizing map: $u:X\to\mathbb{R}^{d_u}$6, $u:X\to\mathbb{R}^{d_u}$7.

Their composition $u:X\to\mathbb{R}^{d_u}$8 describes a nonlinear lifting from the feedback law space $u:X\to\mathbb{R}^{d_u}$9 into the space of operators $\kappa$0. By lifting $\kappa$1 to the RKHS and passing to adjoints, one constructs a linear operator $\kappa$2, $\kappa$3. Identifying $\kappa$4 realizes the bilinear form $\kappa$5, thereby "linearizing" the dependency on both state and policy (Tang, 24 Mar 2025).

3. Functional Analytic Foundations and RKHS–Sobolev Equivalence

Under appropriate regularity assumptions on the system dynamics and kernels—specifically, $\kappa$6 for all $\kappa$7, the determinant of the Jacobian $\kappa$8 bounded away from zero, and choosing Wendland kernel $\kappa$9 on $X$0 with smoothness $X$1—there holds the isomorphism $X$2 (Tang, 24 Mar 2025). This establishes well-posedness for the bounded Koopman operator $X$3 and, by extension, for $X$4.

This guarantees the legitimacy of the feature space embedding, which underpins both theoretical analysis and practical computations, facilitating representation, learning, and error estimation in a functional analytic setting.

4. Data-driven Learning: Kernel EDMD Approximation

Empirical approximation of the Koopman–Nemytskii operator leverages kernel EDMD methodologies. Given data samples $X$5 with $X$6, construct the Gram matrices:

• $X$7,
• $X$8.

By the representer theorem, any Hilbert–Schmidt estimator $X$9 admits a finite-rank expansion:

$\mathcal{N}_\kappa(X)\simeq H^s(X)$0

where $\mathcal{N}_\kappa(X)\simeq H^s(X)$1 minimizes the empirical loss with (or without) regularization.

Approximation modalities include:

• Kernel EDMD (no regularization): $\mathcal{N}_\kappa(X)\simeq H^s(X)$2; $\mathcal{N}_\kappa(X)\simeq H^s(X)$3, $\mathcal{N}_\kappa(X)\simeq H^s(X)$4 projects onto $\mathcal{N}_\kappa(X)\simeq H^s(X)$5.
• Reduced-rank regression (regularized, rank $\mathcal{N}_\kappa(X)\simeq H^s(X)$6): Solve the generalized eigenproblem

$\mathcal{N}_\kappa(X)\simeq H^s(X)$7

extract the $\mathcal{N}_\kappa(X)\simeq H^s(X)$8 leading modes $\mathcal{N}_\kappa(X)\simeq H^s(X)$9, and set $s$0.

This operator learning framework exploits the high expressiveness of RKHSs while maintaining computational tractability through matrix representations.

5. Error Bounds and Predictive Guarantees

The Koopman–Nemytskii operator framework yields explicit, nonasymptotic error bounds for state prediction and cost approximation:

• Generalization for reduced-rank regression (Theorem 4.1): With probability at least $s$1,

$s$2

• Single-step prediction (Thm 4.3): Under an interior-cone condition and mesh fill-distance $s$3,

$s$4

• Multi-step prediction (Thm 4.4): Writing $s$5, $s$6 and similarly for $s$7, then with $s$8,

$s$9

which gives a uniform bound for all $\varpi$0.

• Accumulated cost (Thm 4.5): Let $\varpi$1 (on $\varpi$2) and $\varpi$3 (on $\varpi$4) be kernel-quadratic costs, discount $\varpi$5, with $\varpi$6, $\varpi$7. Then

$\varpi$8

These results provide quantitative control over the performance of data-driven operator approximations for both prediction and cost evaluation tasks.

6. Applications in Nonlinear Closed-Loop Control

Upon learning the Koopman–Nemytskii operator $\varpi$9, several functionalities become available:

• Predict closed-loop trajectories in feature space, then reconstruct state-space quantities via $U$0.
• Approximate cost-to-go $U$1 for any $U$2.
• Optimize control policies using grid search or gradient-based methods in $U$3.

Two application domains are detailed in (Tang, 24 Mar 2025):

• Single-tank level control: For $U$4, $U$5 parameterized by feedback gain $U$6 in $U$7. Utilizing Wendland kernels on $U$8, Gaussian kernel on $U$9, kernel EDMD accurately predicts single-/multi-step states as well as cumulative quadratic cost functions on test grids.
• Williams–Otto reactor: For $\mathcal{N}_\varpi(U)$0, $\mathcal{N}_\varpi(U)$1 parameterized by feedback $\mathcal{N}_\varpi(U)$2 over a specified range. Using $\mathcal{N}_\varpi(U)$3 closed-loop samples, Wendland kernels on $\mathcal{N}_\varpi(U)$4, Gaussian on $\mathcal{N}_\varpi(U)$5, the learned $\mathcal{N}_\varpi(U)$6 achieves low error in state prediction ($\mathcal{N}_\varpi(U)$7 for $\mathcal{N}_\varpi(U)$8) and accumulated cost approximation across policies.

This demonstrates the utility of Koopman–Nemytskii operator learning for both trajectory synthesis and policy optimization in smooth nonlinear systems with state-dependent feedback.

7. Significance and Theoretical Implications

The Koopman–Nemytskii operator framework provides an infinite-dimensional, linear representation of closed-loop, nonlinear controlled dynamics with explicit feedback law dependence. This framework overcomes limitations of prior extensions of the Koopman operator, such as the restriction to autonomous systems or open-loop input dependence only. By employing Sobolev-compatible kernels and leveraging RKHS–Sobolev equivalence, the theory ensures rigorous operator well-posedness and favorable generalization error properties.

A plausible implication is the potential for principled, data-driven control synthesis and evaluation for nonlinear systems in regimes where classical linear or hybrid representations are inadequate. The approach accommodates rich policy classes, exploits regularity of nonlinearities, and provides explicit, nonasymptotic error quantification, setting a foundation for robust nonlinear control in data-centric settings (Tang, 24 Mar 2025).