Abstract
We present a flexible data-driven method for dynamical system analysis that does not require explicit model discovery. The method is rooted in well-established techniques for approximating the Koopman operator from data and is implemented as a semidefinite program that can be solved numerically. Furthermore, the method is agnostic of whether data are generated through a deterministic or stochastic process, so its implementation requires no prior adjustments by the user to accommodate these different scenarios. Rigorous convergence results justify the applicability of the method, while also extending and uniting similar results from across the literature. Examples on discovering Lyapunov functions, performing ergodic optimization, and bounding extrema over attractors for both deterministic and stochastic dynamics exemplify these convergence results and demonstrate the performance of the method.
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Acknowledgements
We are grateful for the hospitality of the University of Surrey during the 2022 ‘Data and Dynamics’ workshop, where this work was started. We also thank Stefan Klus and Enrique Zuazua for their insight into EDMD. JB was partially supported by an Institute of Advanced Studies Fellowship at Surrey and an NSERC Discovery Grant.
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Bramburger, J.J., Fantuzzi, G. Auxiliary Functions as Koopman Observables: Data-Driven Analysis of Dynamical Systems via Polynomial Optimization. J Nonlinear Sci 34, 8 (2024). https://doi.org/10.1007/s00332-023-09990-2
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DOI: https://doi.org/10.1007/s00332-023-09990-2
Keywords
- Koopman operator
- Auxiliary function
- Dynamic mode decomposition
- Semidefinite programming
- Time average
- Lyapunov function