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Koopman Operators for Global Analysis of Hybrid Limit-Cycling Systems: Construction and Spectral Properties

Natsuki Katayama, Yoshihiko Susuki

Abstract

This paper reports a theory of Koopman operators for a class of hybrid dynamical systems with globally asymptotically stable periodic orbits, called hybrid limit-cycling systems. We leverage smooth structures intrinsic to the hybrid dynamical systems, thereby extending the existing theory of Koopman operators for smooth dynamical systems. Rigorous construction of an observable space is carried out to preserve the inherited smooth structures of the hybrid dynamical systems. Complete spectral characterization of the Koopman operators acting on the constructed space is then derived where the existence and uniqueness of their eigenfunctions are ensured. Our results facilitate global analysis of hybrid dynamical systems using the Koopman operator.

Koopman Operators for Global Analysis of Hybrid Limit-Cycling Systems: Construction and Spectral Properties

Abstract

This paper reports a theory of Koopman operators for a class of hybrid dynamical systems with globally asymptotically stable periodic orbits, called hybrid limit-cycling systems. We leverage smooth structures intrinsic to the hybrid dynamical systems, thereby extending the existing theory of Koopman operators for smooth dynamical systems. Rigorous construction of an observable space is carried out to preserve the inherited smooth structures of the hybrid dynamical systems. Complete spectral characterization of the Koopman operators acting on the constructed space is then derived where the existence and uniqueness of their eigenfunctions are ensured. Our results facilitate global analysis of hybrid dynamical systems using the Koopman operator.

Paper Structure

This paper contains 13 sections, 10 theorems, 39 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Hybrid Limit-Cycling Systems
  3. Construction of Koopman Operators
  4. Spectral Properties
  5. An Example
  6. Conclusion
  7. Definition of Hybrid Limit Cycles
  8. Proofs
  9. Preliminaries
  10. Proof of Lemma 1
  11. Proof of Lemma 2
  12. Proof of Theorem 1
  13. Proof of Theorem 2

Key Result

Lemma 1

Let $H$ be a hybrid dynamical system satisfying Assumptions ass:transverseG-ass:finitejump and let $\varphi_t$ be its flow. Then, there exists a triple $n$-dimensional $C^r$ manifold $\tilde{M}$, $C^r$ flow $\{ \tilde{\varphi}_t :\tilde{M} \to \tilde{M}\}_{t>0}$, and $C^r$ surjective map $\pi:M\to \ Furthermore, for all $j\in J$, the restriction ${\pi}|_{M^{(j)}}$ is a ${C}^r$ diffeomorphism.

Figures (4)

  • Figure 1: time-to-impact map $\sigma^{(j)}$ and projection $h^{(j)}$.
  • Figure 2: Construction of a map $\Psi$.
  • Figure 3: An illustrative example.
  • Figure 7: Constructions of the atlas \ref{['eq:atlas']}.

Theorems & Definitions (31)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Remark 1
  • Definition 3
  • Definition 4
  • Lemma 2
  • Definition 5
  • Theorem 1
  • Remark 2
  • ...and 21 more