Data-Driven Control of Linear Parabolic Systems Using Koopman Eigenstructure Assignment

This article 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 determining 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-dynamic mode decomposition (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 to 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.