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A priori error estimate of perturbation method for optimal control problem governed by elliptic PDEs with small uncertainties

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Abstract

In this paper, we investigate the first-order and second-order perturbation approximation schemes for an optimal control problem governed by elliptic PDEs with small uncertainties. The optimal control minimizes the expectation of a cost functional with a deterministic constrained control. First, using a perturbation method, we expand the state and co-state variables up to a certain order with respect to a parameter that controls the magnitude of uncertainty in the input. Then we take the expansions into the known deterministic parametric optimality system to derive the first-order and second-order optimality systems which are both deterministic problems. After that, the two systems are discretized by finite element method directly. The strong and weak error estimates are derived for the state, co-state and control variables, respectively. We finally illustrate the theoretical results by two numerical examples.

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The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions that have helped to improve the quality of this paper.

Funding

This work is supported by the Natural Science Foundation of China (Grant Nos. 11871312, 12131014), the Natural Science Foundation of Shandong Province, China (Grant No. ZR2018MA007) .

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  1. School of Mathematics, Shandong University, Jinan, 250100, Shandong, China

    Mengya Feng & Tongjun Sun

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  1. Mengya Feng
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Correspondence to Tongjun Sun.

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Feng, M., Sun, T. A priori error estimate of perturbation method for optimal control problem governed by elliptic PDEs with small uncertainties. Comput Optim Appl 81, 889–921 (2022). https://doi.org/10.1007/s10589-022-00352-4

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