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Greedy optimal control for elliptic problems and its application to turnpike problems

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Abstract

We adapt and apply greedy methods to approximate in an efficient way the optimal controls for parameterized elliptic control problems. Our results yield an optimal approximation procedure that, in particular, performs better than simply sampling the parameter-space to compute controls for each parameter value. The same method can be adapted for parabolic control problems, but this leads to greedy selections of the realizations of the parameters that depend on the initial datum under consideration. The turnpike property (which ensures that parabolic optimal control problems behave nearly in a static manner when the control horizon is long enough) allows using the elliptic greedy choice of the parameters in the parabolic setting too. We present various numerical experiments and an extensive discussion of the efficiency of our methodology for parabolic control and indicate a number of open problems arising when analyzing the convergence of the proposed algorithms.

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Authors and Affiliations

  1. DeustoTech, University of Deusto, 48007, Bilbao, Basque Country, Spain

    Víctor Hernández-Santamaría & Enrique Zuazua

  2. Facultad de Ingeniería, Universidad de Deusto, Avda. Universidades, 24, 48007, Bilbao, Basque Country, Spain

    Víctor Hernández-Santamaría & Enrique Zuazua

  3. Department of Electrical Engineering and Computing, University of Dubrovnik, Dubrovnik, Croatia

    Martin Lazar

  4. Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain

    Enrique Zuazua

  5. UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, Sorbonne Universités, 75005, Paris, France

    Enrique Zuazua

Authors
  1. Víctor Hernández-Santamaría
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  2. Martin Lazar
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  3. Enrique Zuazua
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Corresponding author

Correspondence to Víctor Hernández-Santamaría.

Additional information

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 694126-DyCon). Part of this research was done while the second author visited DeustoTech and Univesity of Deusto with the support of the DyCon project. The second author was also partially supported by Croatian Science Foundation under ConDyS Project, IP-2016-06-2468. The work of the third author was partially supported by the Grants MTM2014-52347, MTM2017-92996 of MINECO (Spain) and ICON of the French ANR.

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Hernández-Santamaría, V., Lazar, M. & Zuazua, E. Greedy optimal control for elliptic problems and its application to turnpike problems. Numer. Math. 141, 455–493 (2019). https://doi.org/10.1007/s00211-018-1005-z

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  • DOI: https://doi.org/10.1007/s00211-018-1005-z

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