1. Z. Liu, N. Ozay, and E. D. Sontag. Properties of immersions for systems with multiple limit sets with implications to learning Koopman embeddings. . 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.

This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders.

This document was translated from BibT_EX by bibtex2html