Convergence and proposed optimality of adaptive finite element methods for nonlinear optimal control problems

Zuliang Lu; Fei Cai; Ruixiang Xu; Lu Xing; Zuliang Lu; Fei Cai; Ruixiang Xu; Lu Xing

doi:10.3934/math.20221079

AIMS Mathematics

2022, Volume 7, Issue 11: 19664-19695. doi: 10.3934/math.20221079

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Research article

Convergence and proposed optimality of adaptive finite element methods for nonlinear optimal control problems

1.
Key Laboratory for Nonlinear Science and System Structure, Chongqing Three Gorges University, Chongqing 404000, China
2.
Research Center for Mathematics and Economics, Tianjin University of Finance and Economics, Tianjin 300222, China

Received: 26 February 2022 Revised: 04 August 2022 Accepted: 17 August 2022 Published: 06 September 2022
MSC : 49J20, 65N30

This paper investigates the adaptive finite element method for nonlinear optimal control problem, and the research content of reference (^[21] H. Leng and Y. Chen, 2017) is extended accordingly. Linear discretisation of the equation of state and the equation of common state is performed using continuous segmentation functions. At the same time, we use the bubble function technique to prove that the posterior error estimates are obtained from the upper and lower bounds. What is more, for the adaptive finite element method, we also consider convergence and quasi-optimality, where we find that the demand $ h_0\ll 1 $ on the initial grid is unconstrained for the convergence analysis of the proposed adaptive algorithm for the nonlinear optimal control problem. Simultaneously, some numerical simulation is used to verify our theoretical analysis.
- nonlinear optimal control problem,
- adaptive finite element method,
- convergence,
- quasi-optimality
Citation: Zuliang Lu, Fei Cai, Ruixiang Xu, Lu Xing. Convergence and proposed optimality of adaptive finite element methods for nonlinear optimal control problems[J]. AIMS Mathematics, 2022, 7(11): 19664-19695. doi: 10.3934/math.20221079

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Abstract

This paper investigates the adaptive finite element method for nonlinear optimal control problem, and the research content of reference (^[21] H. Leng and Y. Chen, 2017) is extended accordingly. Linear discretisation of the equation of state and the equation of common state is performed using continuous segmentation functions. At the same time, we use the bubble function technique to prove that the posterior error estimates are obtained from the upper and lower bounds. What is more, for the adaptive finite element method, we also consider convergence and quasi-optimality, where we find that the demand $ h_0\ll 1 $ on the initial grid is unconstrained for the convergence analysis of the proposed adaptive algorithm for the nonlinear optimal control problem. Simultaneously, some numerical simulation is used to verify our theoretical analysis.

References

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