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

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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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Authors and Affiliations

  1. Department of Mathematics, PUC-Rio, Rua Marques de São Vicente, 225, Rio de Janeiro, 22453-900, Brazil

    Alessandro Alla

  2. Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, 70569, Stuttgart, Germany

    Bernard Haasdonk & Andreas Schmidt

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  1. Alessandro Alla
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  2. Bernard Haasdonk
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  3. Andreas Schmidt
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Correspondence to Alessandro Alla.

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Communicated by: Anthony Nouy

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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

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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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