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Koopman-Nemytskii Operator: A Linear Representation of Nonlinear Controlled Systems

Wentao Tang

TL;DR

The paper introduces the Koopman‑Nemytskii operator to provide a linear representation of nonlinear controlled dynamics by embedding both states and feedback laws into a product RKHS and mapping to the next‑state feature via $T(\phi_x, \varphi_u)=\phi_{f_u(x)}$. It establishes well‑posedness under Sobolev‑Hilbert/RKHS equivalences, provides a continuous dependence on the policy, and develops kernel EDMD with finite‑rank estimators to perform one‑ and multi‑step state prediction and accumulate control costs. Theoretical results yield $L^{\infty}$ generalization bounds and demonstrate the ability to bound multi‑step errors and costs; practical validation on a liquid storage tank and a 6D chemical reactor confirms accurate state and cost predictions under varying policies. The framework highlights both the potential for data‑driven policy evaluation and the challenges posed by high dimensionality, suggesting future work on stability guarantees and scalable policy synthesis.

Abstract

While Koopman operator lifts a nonlinear system into an infinite-dimensional function space and represents it as a linear dynamics, its definition is restricted to autonomous systems, i.e., does not incorporate inputs or disturbances. To the end of designing state-feedback controllers, the existing extensions of Koopman operator, which only account for the effect of open-loop values of inputs, does not involve feedback laws on closed-loop systems. Hence, in order to generically represent any nonlinear controlled dynamics linearly, this paper proposes a Koopman-Nemytskii operator, defined as a linear mapping from a product reproducing kernel Hilbert space (RKHS) of states and feedback laws to an RKHS of states. Using the equivalence between RKHS and Sobolev-Hilbert spaces under certain regularity conditions on the dynamics and kernel selection, this operator is well-defined. Its data-based approximation, which follows a kernel extended dynamic mode decomposition (kernel EDMD) approach, have established errors in single-step and multi-step state predictions as well as accumulated cost under control.

Koopman-Nemytskii Operator: A Linear Representation of Nonlinear Controlled Systems

TL;DR

The paper introduces the Koopman‑Nemytskii operator to provide a linear representation of nonlinear controlled dynamics by embedding both states and feedback laws into a product RKHS and mapping to the next‑state feature via . It establishes well‑posedness under Sobolev‑Hilbert/RKHS equivalences, provides a continuous dependence on the policy, and develops kernel EDMD with finite‑rank estimators to perform one‑ and multi‑step state prediction and accumulate control costs. Theoretical results yield generalization bounds and demonstrate the ability to bound multi‑step errors and costs; practical validation on a liquid storage tank and a 6D chemical reactor confirms accurate state and cost predictions under varying policies. The framework highlights both the potential for data‑driven policy evaluation and the challenges posed by high dimensionality, suggesting future work on stability guarantees and scalable policy synthesis.

Abstract

While Koopman operator lifts a nonlinear system into an infinite-dimensional function space and represents it as a linear dynamics, its definition is restricted to autonomous systems, i.e., does not incorporate inputs or disturbances. To the end of designing state-feedback controllers, the existing extensions of Koopman operator, which only account for the effect of open-loop values of inputs, does not involve feedback laws on closed-loop systems. Hence, in order to generically represent any nonlinear controlled dynamics linearly, this paper proposes a Koopman-Nemytskii operator, defined as a linear mapping from a product reproducing kernel Hilbert space (RKHS) of states and feedback laws to an RKHS of states. Using the equivalence between RKHS and Sobolev-Hilbert spaces under certain regularity conditions on the dynamics and kernel selection, this operator is well-defined. Its data-based approximation, which follows a kernel extended dynamic mode decomposition (kernel EDMD) approach, have established errors in single-step and multi-step state predictions as well as accumulated cost under control.

Paper Structure

This paper contains 26 sections, 11 theorems, 94 equations, 9 figures.

Table of Contents

  1. Introduction
  2. A motivating example
  3. Contributions of this work
  4. Related works
  5. System identification
  6. Stability considerations in the learning of nonlinear dynamics
  7. Hilbert space and operator formulations in infinite-dimensional systems and stochastic control theory
  8. Data-driven controller synthesis
  9. Organization of the paper and notations
  10. Preliminaries
  11. Hilbert Space and Sobolev-Hilbert Space
  12. Reproducing Kernel Hilbert Space (RKHS)
  13. Koopman Operator
  14. Koopman-Nemytskii Operator
  15. Dependence of the Koopman operator on the feedback law
  16. ...and 11 more sections

Key Result

Proposition 1

At any fixed $u\in\mathcal{C}_{\mathrm{b}}^s(X, \mathbb{R}^{d_a})$, the Nemytskii operator $N_f$ is continuous.

Figures (9)

  • Figure 1: Eigenfunctions of the Koopman operator (associated with eigenvalue $-1$) under different open-loop input values (left) and under different feedback gains (right).
  • Figure 2: Simulation and sampling on the tank model.
  • Figure 3: State prediction error of the Koopman-Nemytskii operator estimated via kernel EDMD.
  • Figure 4: State prediction error of the Koopman-Nemytskii operator estimated via reduced rank regression.
  • Figure 5: Accumulated cost and its prediction under the Koopman-Nemytskii operator.
  • ...and 4 more figures

Theorems & Definitions (31)

  • Definition 1
  • Definition 2
  • Proposition 1
  • proof
  • Definition 3
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Theorem 1
  • ...and 21 more