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Properties of Immersions for Systems with Multiple Limit Sets with Implications to Learning Koopman Embeddings

Zexiang Liu, Necmiye Ozay, Eduardo D. Sontag

Abstract

Linear immersions (such as Koopman eigenfunctions) of a nonlinear system have wide applications in prediction and control. In this work, we study the properties of linear immersions for nonlinear systems with multiple omega-limit sets. While previous research has indicated the possibility of discontinuous one-to-one linear immersions for such systems, it has been unclear whether continuous one-to-one linear immersions are attainable. Under mild conditions, we prove that any continuous immersion to a class of systems including finite-dimensional linear systems collapses all the omega-limit sets, and thus cannot be one-to-one. Furthermore, we show that this property is also shared by approximate linear immersions learned from data as sample size increases and sampling interval decreases. Multiple examples are studied to illustrate our results.

Properties of Immersions for Systems with Multiple Limit Sets with Implications to Learning Koopman Embeddings

Abstract

Linear immersions (such as Koopman eigenfunctions) of a nonlinear system have wide applications in prediction and control. In this work, we study the properties of linear immersions for nonlinear systems with multiple omega-limit sets. While previous research has indicated the possibility of discontinuous one-to-one linear immersions for such systems, it has been unclear whether continuous one-to-one linear immersions are attainable. Under mild conditions, we prove that any continuous immersion to a class of systems including finite-dimensional linear systems collapses all the omega-limit sets, and thus cannot be one-to-one. Furthermore, we show that this property is also shared by approximate linear immersions learned from data as sample size increases and sampling interval decreases. Multiple examples are studied to illustrate our results.
Paper Structure (12 sections, 17 theorems, 45 equations, 6 figures)

This paper contains 12 sections, 17 theorems, 45 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Statement
  4. Technical Definitions
  5. Main Theorem
  6. Implication for Learning Linear Immersions from Data
  7. Extensions of the Main Theorem
  8. Indirect Extensions of Theorem \ref{['main_theorem']}
  9. A Direct Extension of Theorem \ref{['main_theorem']}
  10. Conclusion
  11. Proof of the Main Theorem
  12. Proof of Theorem \ref{['main_theorem_2']}

Key Result

Lemma 1

For any $\xi\in \mathcal{X}$, the $\omega$-limit set $\omega^{+}(\xi)$ is nonempty if the trajectory $\varphi(t,\xi)$ is precompact in $\mathcal{X}$. If the system is linear with $\mathcal{M} = \mathbb{R}^n$ and $\mathcal{X}$ closed, then the converse is also true.

Figures (6)

  • Figure 1: The vector field (red) of the system in \ref{['eq:sys_ex_3']}. The blue curve shows a trajectory of the system that starts from $(0.1,0)$ and converges to the unit circle.
  • Figure 2: For a system $\dot{x} = f(x)$ with three equilibria, the graph of a one-to-one linear immersion (blue) must intersect with the equilibrium set (red) of the immersing linear system at exactly three points.
  • Figure 3: The image of the discontinuous immersion $F$ in \ref{['eqn:F_1d_3']} is shown by the blue lines and dots. The set of equilibria of the immersed system in \ref{['eqn:sys_2d_immersed']} is shown by the red line.
  • Figure 4: The vector field (red) and phase portrait (blue) of the unforced Duffing system in \ref{['eq:sys_duffing']}.
  • Figure 5: The vector field (red) and phase portrait (blue) of the Van der Pol equation in \ref{['eq:van_der_pol']}.
  • ...and 1 more figures

Theorems & Definitions (43)

  • Remark 1
  • Definition 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Example 1
  • Example 2
  • Definition 2
  • Lemma 1
  • ...and 33 more