  In the development of model predictive controllers for PDE-constrained
problems, the use of reduced order models is essential to enable real-time
applicability. Besides local linearization approaches, Proper Orthogonal
Decomposition (POD) has been most widely used in the past in order to derive
such models. Due to the huge advances concerning both theory as well as the
numerical approximation, a very promising alternative based on the Koopman
operator has recently emerged. In this chapter, we present two control
strategies for model predictive control of nonlinear PDEs using data-efficient
approximations of the Koopman operator. In the first one, the dynamic control
system is replaced by a small number of autonomous systems with different yet
constant inputs. The control problem is consequently transformed into a
switching problem. In the second approach, a bilinear surrogate model, is
obtained 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 properties
of these two strategies with respect to solution quality, data requirements,
and complexity of the resulting optimization problem using the 1D Burgers
Equation and the 2D Navier-Stokes Equations as examples. Finally, an extension
for online adaptivity is presented.
