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Koopman operator-based model reduction for switched-system control of PDEs. (English) Zbl 1429.93043

Summary: 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.

MSC:

93B11 System structure simplification
93B28 Operator-theoretic methods
93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
93C20 Control/observation systems governed by partial differential equations
93B52 Feedback control
49J20 Existence theories for optimal control problems involving partial differential equations

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OpenFOAM
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References:

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