Error estimates for spectral approximation of elliptic control problems with integral state and control constraints

The goal of this paper is to investigate the Legendre–Galerkin spectral approximation of elliptic optimal control problems with integral state and control constraints. Thanks to the appropriate base functions of the discrete spaces, the discrete system is with sparse coefficient matrices. We first present the optimality conditions of the control system. Then a priori and a posteriori error estimates both in H1H1 and L2L2 norms are derived. Some numerical tests indicate that the spectral accuracy can be achieved, and the proposed method is competitive for solving control problems.