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Koopman Operator in the Weighted Function Spaces and its Learning for the Estimation of Lyapunov and Zubov Functions

Wentao Tang

TL;DR

By defining the Koopman operator on a space of weighted continuous functions and learning it on a weighted reproducing kernel Hilbert space, the Koopman operator is guaranteed to be contractive and the accumulated error is bounded.

Abstract

The mathematical properties and data-driven learning of the Koopman operator, which represents nonlinear dynamics as a linear mapping on a properly defined functional spaces, have become key problems in nonlinear system identification and control. However, Koopman operators that are approximately learned from snapshot data may not always accurately predict the system evolution on long horizons. In this work, by defining the Koopman operator on a space of weighted continuous functions and learning it on a weighted reproducing kernel Hilbert space, the Koopman operator is guaranteed to be contractive and the accumulation learning error is bounded. The weighting function, assumed to be known a priori, has an exponential decay with the flow or decays exponentially when compensated by an exponential factor. Under such a construction, the Koopman operator learned from data is used to estimate (i) Lyapunov functions for globally asymptotically stable dynamics, and (ii) Zubov-Lyapunov functions that characterize the domain of attraction. For these estimations, probabilistic bounds on the errors are derived.

Koopman Operator in the Weighted Function Spaces and its Learning for the Estimation of Lyapunov and Zubov Functions

TL;DR

By defining the Koopman operator on a space of weighted continuous functions and learning it on a weighted reproducing kernel Hilbert space, the Koopman operator is guaranteed to be contractive and the accumulated error is bounded.

Abstract

The mathematical properties and data-driven learning of the Koopman operator, which represents nonlinear dynamics as a linear mapping on a properly defined functional spaces, have become key problems in nonlinear system identification and control. However, Koopman operators that are approximately learned from snapshot data may not always accurately predict the system evolution on long horizons. In this work, by defining the Koopman operator on a space of weighted continuous functions and learning it on a weighted reproducing kernel Hilbert space, the Koopman operator is guaranteed to be contractive and the accumulation learning error is bounded. The weighting function, assumed to be known a priori, has an exponential decay with the flow or decays exponentially when compensated by an exponential factor. Under such a construction, the Koopman operator learned from data is used to estimate (i) Lyapunov functions for globally asymptotically stable dynamics, and (ii) Zubov-Lyapunov functions that characterize the domain of attraction. For these estimations, probabilistic bounds on the errors are derived.
Paper Structure (13 sections, 12 theorems, 51 equations, 3 figures)

This paper contains 13 sections, 12 theorems, 51 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Koopman Operator for Nonlinear Dynamics
  4. Reproducing Kernel Hilbert Space (RKHS)
  5. Approximately Learning Koopman Operators on RKHS
  6. Koopman and Zubov-Koopman Operators in Weighted Function Spaces
  7. Koopman Operator in Weighted Function Spaces
  8. Learning Error of Koopman Operators in Weighted RKHS
  9. Extension to Zubov-Koopman Operators
  10. Estimation of Lyapunov and Zubov Functions
  11. Estimation of Lyapunov Function
  12. Estimation of Zubov Function and DoA
  13. Conclusions

Key Result

Proposition 1

The estimated Koopman operator $\hat{A}_{\beta, r}$ obtained above satisfies the following bound: where $\lambda_{\max}(L)$ represents the largest eigenvalue of $L$.

Figures (3)

  • Figure 1: Prediction of Lyapunov function by the Koopman operator learned on weighted RKHS.
  • Figure 2: Prediction of quadratic functions by the Koopman operator learned on weighted RKHS.
  • Figure 3: Prediction of the Zubov function by the Zubov-Koopman operator learned on weighted RKHS.

Theorems & Definitions (27)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • proof
  • Definition 4
  • Proposition 2
  • Definition 5
  • Proposition 3
  • proof
  • ...and 17 more