Properties of Immersions for Systems with Multiple Limit Sets with Implications to Learning Koopman Embeddings
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913492512014336 |
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| author | Liu, Zexiang Ozay, Necmiye Sontag, Eduardo D. |
| author_facet | Liu, Zexiang Ozay, Necmiye Sontag, Eduardo D. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_17045 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Properties of Immersions for Systems with Multiple Limit Sets with Implications to Learning Koopman Embeddings Liu, Zexiang Ozay, Necmiye Sontag, Eduardo D. Systems and Control Dynamical Systems 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. |
| title | Properties of Immersions for Systems with Multiple Limit Sets with Implications to Learning Koopman Embeddings |
| topic | Systems and Control Dynamical Systems |
| url | https://arxiv.org/abs/2312.17045 |