Communications in Mathematical Sciences
Volume 20 (2022)
Error analysis of Galerkin spectral methods for nonlinear optimal control problems with integral control constraint
Pages: 1659 – 1683
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.
nonlinear optimal control problem, control constraint, Galerkin spectral methods, a priori error analysis, a posteriori error analysis
2010 Mathematics Subject Classification
49J20, 65K10, 65N35
This work is supported by the State Key Program of National Natural Science Foundation of China (11931003) and National Natural Science Foundation of China (41974133, 11971410).
Received 8 June 2021
Received revised 12 December 2021
Accepted 17 January 2022
Published 14 September 2022