Abstract
In this chapter, we present a new framework to study global stability of nonlinear systems. The proposed approach is based on the stability properties of the Koopman operator and can be seen as an extension of classic stability analysis of linear systems. In the case of (hyperbolic) equilibria, we show that the existence of specific eigenfunctions of the operator is a necessary and sufficient condition for global stability of the attractor. Moreover, using the realization of the operator in a finite-dimensional basis, we provide a systematic method to compute candidate Lyapunov functions of stable systems.
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Notes
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A sum-of-squares relaxation for polynomial optimization problem is technically a restriction since the space of possible solutions is smaller. However, this nomenclature became standard and the relaxation should be understood as making the problem tractable.
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Acknowledgements
Dr. Sootla was supported by the EPSRC grant EP/M002454/1. I. Mezić acknowledges support from ARO-MURI grant W911NF-17-1-0306.
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Mauroy, A., Sootla, A., Mezić, I. (2020). Koopman Framework for Global Stability Analysis. In: Mauroy, A., Mezić, I., Susuki, Y. (eds) The Koopman Operator in Systems and Control. Lecture Notes in Control and Information Sciences, vol 484. Springer, Cham. https://doi.org/10.1007/978-3-030-35713-9_2
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