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

TL;DR

This paper introduces a Koopman operator-based reduced order modeling framework for efficient optimal and feedback control of PDEs, transforming complex nonlinear problems into low-dimensional linear systems for faster computation.

Contribution

The paper develops a novel approach using Koopman operator approximations to enable fast, accurate control of PDEs by converting nonlinear control problems into linear switching time optimization problems.

Findings

Significant reduction in solution times for PDE control problems.

High accuracy of control solutions using Koopman-based models.

Successful application to Burgers and Navier--Stokes equations.

Abstract

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. In situations where the Koopman operator can be computed exactly using Extended Dynamic Mode Decomposition (EDMD), the proposed approach yields optimal control inputs. Furthermore, a recent convergence result for EDMD suggests that the approach can be applied to more complex dynamics as well. 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.