## Learning Koopman eigenfunctions for prediction and control: the transient case

#### Milan KordaCentre National de la Recherche Scientifique (CNRS)

This talk will present a data-driven framework for learning eigenfunctions of the Koopman operator geared toward prediction and control. The method leverages the richness of the spectrum of the Koopman operator in the transient, off-attractor, regime to construct a large number of eigenfunctions such that the function whose values we wish to predict lies in their span and hence is predictable in a linear fashion. Once a predictor for the uncontrolled part of the system is obtained in this way, the incorporation of control is done through a multi-step prediction error minimization. The predictor so obtained is in the form of a linear dynamical system and can be readily applied within a Koopman model predictive control framework to control nonlinear dynamical systems using linear model predictive control tools. The method is entirely data-driven and based on convex optimization. The eigenfunction construction method is also analyzed theoretically, proving that the family of eigenfunctions obtained is dense in the space of continuous functions. In addition, the method is extended to construct generalized eigenfunctions that also give rise Koopman invariant subspaces and hence can be used for linear prediction.

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