Galerkin spectral approximation of optimal control problems with L2-norm control constraint

In this paper, an optimal control problem governed by elliptic partial differential equations, whose objective functionals do not depend on the controls, with L-2-norm constraint on control variable is considered, and Galerkin spectral approximation of the optimal control problem without penalty term is established. To analyze the optimal control problem, optimality conditions of the control problem are first derived in detail. By virtue of some lemmas and auxiliary equations, a priori error estimates of spectral approximation for optimal control problems are analyzed rigorously. Moreover, a posteriori error estimates of the optimal control problem are also established carefully. In the end, some numerical examples are carried out to confirm the analytical results that the errors decay exponentially fast as long as the data is smooth. (C) 2019 IMACS. Published by Elsevier B.V. All rights reserved.