Error estimates of variational discretization for semilinear parabolic optimal control problems

Chunjuan Hou; Zuliang Lu; Xuejiao Chen; Fei Huang; Chunjuan Hou; Zuliang Lu; Xuejiao Chen; Fei Huang

doi:10.3934/math.2021047

AIMS Mathematics

2021, Volume 6, Issue 1: 772-793. doi: 10.3934/math.2021047

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Research article

Error estimates of variational discretization for semilinear parabolic optimal control problems

1.
Huashang College Guangdong University of Finance and Economics, Guangzhou, 511300, P. R. China
2.
Key Laboratory for Nonlinear Science and System Structure, Chongqing Three Gorges University, Chongqing, 404000, P. R. China
3.
Research Center for Mathematics and Economics, Tianjin University of Finance and Economics, Tianjin, 300222, P. R. China

Received: 14 May 2020 Accepted: 09 July 2020 Published: 02 November 2020
MSC : 49J20, 65N30

In this paper, variational discretization directed against the optimal control problem governed by nonlinear parabolic equations with control constraints is studied. It is known that the a priori error estimates is $|||u-u_h|||_{L^\infty(J; L^2(\Omega))} = O(h+k)$ using backward Euler method for standard finite element. In this paper, the better result $|||u-u_h|||_{L^\infty(J; L^2(\Omega))} = O(h^2+k)$ is gained. Beyond that, we get a posteriori error estimates of residual type.
- optimal control problems,
- semilinear parabolic equations,
- error estimates,
- finite element methods
Citation: Chunjuan Hou, Zuliang Lu, Xuejiao Chen, Fei Huang. Error estimates of variational discretization for semilinear parabolic optimal control problems[J]. AIMS Mathematics, 2021, 6(1): 772-793. doi: 10.3934/math.2021047

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Abstract

In this paper, variational discretization directed against the optimal control problem governed by nonlinear parabolic equations with control constraints is studied. It is known that the a priori error estimates is $|||u-u_h|||_{L^\infty(J; L^2(\Omega))} = O(h+k)$ using backward Euler method for standard finite element. In this paper, the better result $|||u-u_h|||_{L^\infty(J; L^2(\Omega))} = O(h^2+k)$ is gained. Beyond that, we get a posteriori error estimates of residual type.

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