Feedback control of parametrized PDEs via model order reduction and dynamic programming principle

Abstract

In this paper, we investigate infinite horizon optimal control problems for parametrized partial differential equations. We are interested in feedback control via dynamic programming equations which is well-known to suffer from the curse of dimensionality. Thus, we apply parametric model order reduction techniques to construct low-dimensional subspaces with suitable information on the control problem, where the dynamic programming equations can be approximated. To guarantee a low number of basis functions, we combine recent basis generation methods and parameter partitioning techniques. Furthermore, we present a novel technique to construct non-uniform grids in the reduced domain, which is based on statistical information. Finally, we discuss numerical examples to illustrate the effectiveness of the proposed methods for PDEs in two space dimensions.

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Acknowledgments

The authors would like to thank Max Gunzburger for several fruitful discussions on the subject and valuable comments for improvement of the presentation.

Funding

The first author was supported by US Department of Energy grant number DE-SC0009324. The second and third authors would like to thank the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/2) at the University of Stuttgart.

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Correspondence to Alessandro Alla.

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This article belongs to the Topical Collection: Model Reduction of Parametrized Systems

Guest Editors: Anthony Nouy, Peter Benner, Mario Ohlberger, Gianluigi Rozza, Karsten Urban and Karen Willcox

Communicated by: Anthony Nouy

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Alla, A., Haasdonk, B. & Schmidt, A. Feedback control of parametrized PDEs via model order reduction and dynamic programming principle. Adv Comput Math 46, 9 (2020). https://doi.org/10.1007/s10444-020-09744-8

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Keywords

  • Dynamic programming principle
  • Semi-Lagrangian schemes
  • Hamilton-Jacobi-Bellman equations
  • Optimal control
  • Model reduction
  • Reduced basis method

Mathematics Subject Classification (2010)

  • 49L20
  • 49J20
  • 65N99
  • 78M34