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Optimal actuator location for time and norm optimal control of null controllable heat equation

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Abstract

We consider minimal time and minimal norm of the optimal controls for a multi-dimensional internally controlled heat equation with control domain varying over a class of open sets. Two problems are formulated separately into different types of shape optimization problems over this open set class. The governing equation with any given initial value and admissible control are considered as constraints, and minimal time or minimal norm of the optimal controls is considered as cost for related shape optimization. The solution of the shape optimization leads to the optimal actuator location for optimal controls. The existence of such an optimal location domain for both minimal time and minimal norm controls is presented. A different problem that relates the balance between minimal time and minimal norm controls is also discussed. This study builds a link between optimal control and shape optimization for this multi-dimensional heat equation.

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Acknowledgments

The authors are grateful to Professor Lijuan Wang of Wuhan University for the helpful discussions about the problems discussed in this paper. The anonymous reviewers indicated that the proof of Lemma 7 is similar to related part in reference [19] that is unfortunately not available in the English world.

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Authors and Affiliations

  1. Academy of Mathematics and Systems Science, Academia Sinica, Beijing , 100190, People’s Republic of China

    Bao-Zhu Guo

  2. School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa

    Bao-Zhu Guo

  3. Department of Mathematics, King Abdulaziz University, Jeddah , 21589, Saudi Arabia

    Bao-Zhu Guo

  4. School of Mathematics and Statistics, Central South University, Changsha , 410075, People’s Republic of China

    Dong-Hui Yang

Authors
  1. Bao-Zhu Guo
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  2. Dong-Hui Yang
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Correspondence to Dong-Hui Yang.

Additional information

This work was carried out with the support of the National Natural Science Foundation of China, the National Basic Research Program of China (2011CB808002), and the National Research Foundation of South Africa.

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Guo, BZ., Yang, DH. Optimal actuator location for time and norm optimal control of null controllable heat equation. Math. Control Signals Syst. 27, 23–48 (2015). https://doi.org/10.1007/s00498-014-0133-y

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  • DOI: https://doi.org/10.1007/s00498-014-0133-y

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