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Finite Element Approximation of Optimal Control Problem Governed by Space Fractional Equation

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Abstract

In this paper we investigate finite element approximation of optimal control problem governed by space fractional diffusion equation with control constraints. The control variable is approximated by piecewise constant. Regularity estimate for the control problem is proved based on the first order optimality system and a priori error estimates for the state, the adjoint state and the control variables are derived. Due to the nonlocal property of fractional derivative, which will leads to a full stiff matrix, we develop a fast primal dual active set algorithm for the control problem. Numerical examples are given to illustrate the theoretical findings and the efficiency of the fast algorithm.

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Acknowledgements

The authors would like to thank the referees for their careful reviews and many valuable suggestions which have led to a considerably improved paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, Jinan, China

    Zhaojie Zhou

  2. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong

    Zhiyu Tan

Authors
  1. Zhaojie Zhou
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  2. Zhiyu Tan
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Correspondence to Zhiyu Tan.

Additional information

The research was supported by Natural Science Foundation of Shandong Province (No. ZR2016JL004, ZR2017MA020) and National Natural Science Foundation of China (No. 11301311, 11471196).

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Zhou, Z., Tan, Z. Finite Element Approximation of Optimal Control Problem Governed by Space Fractional Equation. J Sci Comput 78, 1840–1861 (2019). https://doi.org/10.1007/s10915-018-0829-0

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  • DOI: https://doi.org/10.1007/s10915-018-0829-0

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