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

  • Víctor Hernández-Santamaría
  • Martin Lazar
  • Enrique Zuazua
Article
  • 24 Downloads

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.

Mathematics Subject Classification

49J20 49K20 93C20 49N05 65K10 

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Víctor Hernández-Santamaría
    • 1
    • 2
  • Martin Lazar
    • 3
  • Enrique Zuazua
    • 1
    • 2
    • 4
    • 5
  1. 1.DeustoTech, University of DeustoBilbaoSpain
  2. 2.Facultad de IngenieríaUniversidad de DeustoBilbaoSpain
  3. 3.Department of Electrical Engineering and ComputingUniversity of DubrovnikDubrovnikCroatia
  4. 4.Departamento de MatemáticasUniversidad Autónoma de MadridMadridSpain
  5. 5.UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis LionsSorbonne UniversitésParisFrance

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