  This paper considers the data-driven stabilization of linear boundary
controlled parabolic PDEs by making use of the Koopman operator. For this, a
Koopman eigenstructure assignment problem is solved, which amounts to determine
a feedback of the Koopman open-loop eigenfunctionals assigning a desired finite
set of closed-loop Koopman eigenvalues and eigenfunctionals to the closed-loop
system. It is shown that the designed controller only needs a finite number of
open-loop Koopman eigenvalues and modes of the state. They are determined by
extending the classical Krylov-DMD to parabolic systems. For this, only a
finite number of pointlike outputs and their temporal samples as well as
temporal samples of the inputs are required resulting in a data-driven solution
of the eigenstructure assignment problem. Exponential stability of the
closed-loop system in the presence of small Krylov-DMD errors is verified. An
unstable diffusion-reaction system demonstrates the new data-driven controller
design technique for distributed-parameter systems.
