A posteriori error estimates of spectral method for nonlinear parabolic optimal control problem

Lin Li; Zuliang Lu; Wei Zhang; Fei Huang; Yin Yang

doi:10.1186/s13660-018-1729-4

A posteriori error estimates of spectral method for nonlinear parabolic optimal control problem

J Inequal Appl. 2018;2018(1):138. doi: 10.1186/s13660-018-1729-4. Epub 2018 Jun 19.

Authors

Lin Li¹, Zuliang Lu^{1

2

3}, Wei Zhang⁴, Fei Huang¹, Yin Yang^{5

6}

Affiliations

¹ 1Key Laboratory for Nonlinear Science and System Structure, Chongqing Three Gorges University, Chongqing, P.R. China.
² 2Key Laboratory of Intelligent Information Processing and Control, Chongqing Three Gorges University, Chongqing, P.R. China.
³ 3Research Center for Mathematics and Economics, Tianjin University of Finance and Economics, Tianjin, P.R. China.
⁴ 4Key Laboratory of Intelligent Information Processing and Control of Chongqing Municipal Institutions of Higher Education, Chongqing Three Gorges University, Chongqing, P.R. China.
⁵ 5Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, P.R. China.
⁶ 6Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, Xiangtan University, Xiangtan, P.R. China.

Abstract

In this paper, we investigate the spectral approximation of optimal control problem governed by nonlinear parabolic equations. A spectral approximation scheme for the nonlinear parabolic optimal control problem is presented. We construct a fully discrete spectral approximation scheme by using the backward Euler scheme in time. Moreover, by using an orthogonal projection operator, we obtain $L^{2} (H^{1}) - L^{2} (L^{2})$ a posteriori error estimates of the approximation solutions for both the state and the control. Finally, by introducing two auxiliary equations, we also obtain $L^{2} (L^{2}) - L^{2} (L^{2})$ a posteriori error estimates of the approximation solutions for both the state and the control.

Keywords: A posteriori error estimates; Nonlinear parabolic equations; Optimal control problem; Spectral method; Variational discretization.