Communications in Mathematical Sciences

Volume 20 (2022)

Number 6

Error analysis of Galerkin spectral methods for nonlinear optimal control problems with integral control constraint

Pages: 1659 – 1683

DOI: https://dx.doi.org/10.4310/CMS.2022.v20.n6.a9

Authors

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

Xiuxiu Lin (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

The error analysis of Galerkin spectral methods for integral control constrained nonlinear optimal control problems is investigated in this paper. At first, the optimality conditions of the optimal control problem are presented. More precisely, on the basis of the property of projection operator, a priori error analysis of Galerkin spectral discretization is derived. Moreover, a posteriori error analysis of state, control, adjoint state is established rigorously. Furthermore, for this nonlinear problem, detailed a posteriori error analysis of $hp$ spectral element discretization for the optimal control problem is also proved. In the end, ample numerical experiments are presented to verify the theoretical analysis of Galerkin spectral discretization by using the efficient gradient projection algorithm.

Keywords

nonlinear optimal control problem, control constraint, Galerkin spectral methods, a priori error analysis, a posteriori error analysis

2010 Mathematics Subject Classification

49J20, 65K10, 65N35

Received 8 June 2021

Received revised 12 December 2021

Accepted 17 January 2022

Published 14 September 2022