期刊
AUTOMATICA
卷 106, 期 -, 页码 184-191出版社
PERGAMON-ELSEVIER SCIENCE LTD
DOI: 10.1016/j.automatica.2019.05.016
关键词
Koopman operator; Dynamic mode decomposition; Reduced order modeling; Optimal control; Switched systems
资金
- DFG Priority Programme 1962 Non-smooth and Complementarity-based Distributed Parameter Systems
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. (C) 2019 Elsevier Ltd. All rights reserved.
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