1. Z. Liu, N. Ozay, and E. D. Sontag. Properties of immersions for systems with multiple limit sets with implications to learning Koopman embeddings. 2024. Note: Submitted. Preprint in arXiv:2312.18045v1, 28 Dec 2023.Keyword(s): linear systems, nonlinear systems, observables, Koopman embedding, duality. Abstract:

   Linear immersions (or Koopman eigenmappings) of a nonlinear system have wide applications in prediction and control. In this work, we study the non-existence of one-to-one 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 remained uncertain whether continuous one-to-one linear immersions are attainable. Under mild conditions, we prove that any continuous one-to-one immersion to a class of systems including linear systems cannot distinguish different 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.

1. Z. Liu, N. Ozay, and E. D. Sontag. On the non-existence of immersions for systems with multiple omega-limit sets. In 22nd IFAC World Congress, IFAC-PapersOnLine, volume 56, pages 60-64, 2023. Note: This is a preliminary version of the journal paper Properties of immersions for systems with multiple limit sets with implications to learning Koopman embeddings.[PDF] [doi:https://doi.org/10.1016/j.ifacol.2023.10.1408] Keyword(s): linear systems, nonlinear systems, observables, Koopman embedding, duality. Abstract:

   Linear immersions (or Koopman eigenmappings) of a nonlinear system have wide applications in prediction and control. In this work, we study the existence of one-to-one linear immersions for nonlinear systems with multiple omega-limit sets. For this class of systems, existing work shows that a discontinuous one-to-one linear immersion may exist, but it is unclear if a continuous one-to-one linear immersion exists. Under mild conditions, we prove that systems with multiple omega-limit sets cannot admit a continuous one-to-one immersion to a class of systems including linear systems.

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