Koopman operator-based model reduction for switched-system control of PDEs

Sebastian Peitz*, Stefan Klus

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

100 Citations (Scopus)


We present a new framework for optimal and feedback control of PDEs using Koopman operator-based reduced order models (K-ROMs). The Koopman operator is a linear but infinite-dimensional operator which describes the dynamics of observables. A numerical approximation of the Koopman operator therefore yields a linear system for the observation of an autonomous dynamical system. In our approach, by introducing a finite number of constant controls, the dynamic control system is transformed into a set of autonomous systems and the corresponding optimal control problem into a switching time optimization problem. This allows us to replace each of these systems by a K-ROM which can be solved orders of magnitude faster. By this approach, a nonlinear infinite-dimensional control problem is transformed into a low-dimensional linear problem. Using a recent convergence result for the numerical approximation via Extended Dynamic Mode Decomposition (EDMD), we show that the value of the K-ROM based objective function converges in measure to the value of the full objective function. To illustrate the results, we consider the 1D Burgers equation and the 2D Navier–Stokes equations. The numerical experiments show remarkable performance concerning both solution times and accuracy.

Original languageEnglish
Pages (from-to)184-191
Number of pages8
Publication statusPublished - Aug 2019


  • Dynamic mode decomposition
  • Koopman operator
  • Optimal control
  • Reduced order modeling
  • Switched systems

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Electrical and Electronic Engineering


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