In recent years, the success of the Koopman operator in dynamical systems analysis has also fueled the development of Koopman operator-based control frameworks. In order to preserve the relatively low data requirements for an approximation via Dynamic Mode Decomposition, a quantization approach was recently proposed in [Peitz & Klus, Automatica 106, 2019]. This way, control of nonlinear dynamical systems can be realized by means of switched systems techniques, using only a finite set of autonomous Koopman operator-based reduced models. These individual systems can be approximated very efficiently from data. The main idea is to transform a control system into a set of autonomous systems for which the optimal switching sequence has to be computed. In this article, we extend these results to continuous control inputs using relaxation. This way, we combine the advantages of the data efficiency of approximating a finite set of autonomous systems with continuous controls. We show that when using the Koopman generator, this relaxation --- realized by linear interpolation between two operators --- does not introduce any error for control affine systems. This allows us to control high-dimensional nonlinear systems using bilinear, low-dimensional surrogate models. The efficiency of the proposed approach is demonstrated using several examples with increasing complexity, from the Duffing oscillator to the chaotic fluidic pinball.
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