TY - JOUR
T1 - GALERKIN SPECTRAL METHOD FOR ELLIPTIC OPTIMAL CONTROL PROBLEM WITH L2-NORM CONTROL CONSTRAINT
AU - Tao, Zhen Zhen
AU - Sun, Bing
N1 - Publisher Copyright:
© 2022 American Institute of Mathematical Sciences. All rights reserved.
PY - 2022/8
Y1 - 2022/8
N2 - This paper is concerned with the Galerkin spectral approximation of an optimal control problem governed by the elliptic partial differential equations (PDEs). Its objective functional depends on the control variable governed by the L2-norm constraint. The optimality conditions for both the optimal control problem and its corresponding spectral approximation problem are given, successively. Thanks to some lemmas and the auxiliary systems, a priori error estimates of the Galerkin spectral approximation problem are established in detail. Moreover, a posteriori error estimates of the spectral approximation problem are also investigated, which include not only H1-norm error for the state and co-state but also L2-norm error for the control, state and costate. Finally, three numerical examples are executed to demonstrate the errors decay exponentially fast.
AB - This paper is concerned with the Galerkin spectral approximation of an optimal control problem governed by the elliptic partial differential equations (PDEs). Its objective functional depends on the control variable governed by the L2-norm constraint. The optimality conditions for both the optimal control problem and its corresponding spectral approximation problem are given, successively. Thanks to some lemmas and the auxiliary systems, a priori error estimates of the Galerkin spectral approximation problem are established in detail. Moreover, a posteriori error estimates of the spectral approximation problem are also investigated, which include not only H1-norm error for the state and co-state but also L2-norm error for the control, state and costate. Finally, three numerical examples are executed to demonstrate the errors decay exponentially fast.
KW - Galerkin spectral approximation
KW - a posteriori error estimates
KW - a priori error estimates
KW - optimal control
KW - optimality conditions
UR - http://www.scopus.com/inward/record.url?scp=85134558692&partnerID=8YFLogxK
U2 - 10.3934/dcdsb.2021220
DO - 10.3934/dcdsb.2021220
M3 - Article
AN - SCOPUS:85134558692
SN - 1531-3492
VL - 27
SP - 4121
EP - 4141
JO - Discrete and Continuous Dynamical Systems - Series B
JF - Discrete and Continuous Dynamical Systems - Series B
IS - 8
ER -