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In the development of model predictive controllers for PDE-constrainedproblems, the use of reduced order models is essential to enable real-timeapplicability. Besides local linearization approaches, Proper OrthogonalDecomposition (POD) has been most widely used in the past in order to derivesuch models. Due to the huge advances concerning both theory as well as thenumerical approximation, a very promising alternative based on the Koopmanoperator has recently emerged. In this chapter, we present two controlstrategies for model predictive control of nonlinear PDEs using data-efficientapproximations of the Koopman operator. In the first one, the dynamic controlsystem is replaced by a small number of autonomous systems with different yetconstant inputs. The control problem is consequently transformed into aswitching problem. In the second approach, a bilinear surrogate model, isobtained via linear interpolation between two of these autonomous systems.Using a recent convergence result for Extended Dynamic Mode Decomposition(EDMD), convergence to the true optimum can be proved. We study the propertiesof these two strategies with respect to solution quality, data requirements,and complexity of the resulting optimization problem using the 1D BurgersEquation and the 2D Navier-Stokes Equations as examples. Finally, an extensionfor online adaptivity is presented.

Feedback control of nonlinear PDEs using data-efficient reduced order models based on the Koopman operator