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Error Estimates of hp Spectral Element Methods in Nonlinear Optimal Control Problem

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Abstract

The main purpose of this paper is to discuss hp spectral element method for optimal control problem governed by a nonlinear elliptic equation with \(L^2\)-norm constraint for control variable. We then set up its weak formulation and hp spectral element approximation scheme. A priori error estimates of hp spectral element approximation based on some suitable projection operators are proved carefully. Using some properties of projection operators, a posteriori error estimates for both the state and the control approximation under some reasonable assumptions are established rigorously. Such estimates are useful tools, which can be used to construct reliable adaptive spectral element methods for optimal control problems.

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Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

The authors wish to thank the editor and referees for their helpful and constructive comments that improved the presentation. The work of Xiuxiu Lin is supported by National Natural Science Foundation of China (No. 12201222) and the Scientific Research and Cultivation Fund for Young Teachers of South China Normal University(No. 21KJ17). The work of Yanping Chen is supported by the State Key Program of National Natural Science Foundation of China (No. 11931003) and National Natural Science Foundation of China (No. 41974133). The work of Yunqing Huang is supported by National Natural Science Foundation of China (No. 11971410).

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Authors and Affiliations

  1. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, People’s Republic of China

    Xiuxiu Lin & Yanping Chen

  2. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, People’s Republic of China

    Yunqing Huang

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  1. Xiuxiu Lin
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  2. Yanping Chen
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Correspondence to Yanping Chen.

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Communicated by Anthony Bloch.

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The work of Xiuxiu Lin is supported by National Natural Science Foundation of China (No. 12201222) and the Scientific Research and Cultivation Fund for Young Teachers of South China Normal University(No. 21KJ17). The work of Yanping Chen is supported by the State Key Program of National Natural Science Foundation of China (No. 11931003) and National Natural Science Foundation of China (No. 41974133). The work of Yunqing Huang is supported by National Natural Science Foundation of China (No. 11971410).

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Lin, X., Chen, Y. & Huang, Y. Error Estimates of hp Spectral Element Methods in Nonlinear Optimal Control Problem. J Nonlinear Sci 34, 21 (2024). https://doi.org/10.1007/s00332-023-09991-1

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