Communications in Mathematical Sciences

Volume 19 (2021)

Number 5

Galerkin spectral methods for an elliptic optimal control problem with $L^2$-norm state constraint

Pages: 1247 – 1267

DOI: https://dx.doi.org/10.4310/CMS.2021.v19.n5.a4

Authors

Xiuxiu Lin (School of Mathematical Sciences, South China Normal University, Guangzhou, China)

Yanping Chen (School of Mathematical Sciences, South China Normal University, Guangzhou, China)

Yunqing Huang (Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan, China)

Abstract

In this paper, an optimal control problem governed by elliptic equations with $L^2$‑norm constraint for state variable is developed. Firstly, the optimality conditions of the optimal control problem are derived, and the optimal control problem is approximated by the Galerkin spectral methods. Similarly, the optimality conditions of the discrete problem are also obtained. Then, some important lemmas are proved to obtain a priori error estimates of the coupled state and control approximation rigorously. Moreover, a posteriori error estimates are also established for the optimal control problem carefully. Finally, based on the projection gradient algorithm, some numerical experiments are presented to confirm our analytical findings. It is proved that the exponential convergence rate can be achieved.

Keywords

optimal control problems, $L^2$-norm state constraint, Galerkin spectral methods, a priori error estimates, a posteriori error estimates

2010 Mathematics Subject Classification

34K28, 49J20, 65K10, 65N35

This work is supported by the State Key Program of National Natural Science Foundation of China (11931003), and by the National Natural Science Foundation of China (41974133, 11671157,11971410).

Received 26 February 2019

Accepted 30 December 2020

Published 7 July 2021