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The Koopman operator theory provides a global linearization framework for general nonlinear dynamics, offering significant advantages for system analysis and control. However, practical applications typically involve approximating the infinite-dimensional Koopman operator in a lifted space spanned by a finite set of observable functions. The accuracy of this approximation is the key to effective Koopman operator-based analysis and control methods, generally improving as the dimension of the observables increases. Nonetheless, this increase in dimensionality significantly escalates both storage requirements and computational complexity, particularly for high-dimensional systems, thereby limiting the applicability of these methods in real-world problems. In this paper, we address this problem by reformulating the Koopman operator in tensor format to break the curse of dimensionality associated with its approximation through tensor decomposition techniques. This effective reduction in complexity enables the selection of high-dimensional observable functions and the handling of large-scale datasets, which leads to a precise linear prediction model utilizing the tensor-based Koopman operator. Furthermore, we propose an optimal control framework with the tensor-based Koopman operator, which adeptly addresses the nonlinear dynamics and constraints by linear reformulation in the lifted space and significantly reduces the computational complexity through separated representation of the tensor structure.
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Is the Finite-Time Lyapunov Exponent Field a Koopman Eigenfunction?
This work serves as a bridge between two approaches to analysis of dynamical systems: the local, geometric analysis, and the global operator theoretic Koopman analysis. We explicitly construct vector fields where the instantaneous Lyapunov exponent field is a Koopman eigenfunction. Restricting ourselves to polynomial vector fields to make this construction easier, we find that such vector fields do exist, and we explore whether such vector fields have a special structure, thus making a link between the geometric theory and the transfer operator theory.
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- PAR ID:
- 10301681
- Date Published:
- Journal Name:
- Mathematics
- Volume:
- 9
- Issue:
- 21
- ISSN:
- 2227-7390
- Page Range / eLocation ID:
- 2731
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3390/math9212731
