GALERKIN SPECTRAL METHODS FOR AN ELLIPTIC OPTIMAL CONTROL PROBLEM WITH L2-NORM STATE CONSTRAINT

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.