Legendre-Galerkin spectral methods for optimal control problems with integral constraint for state in one dimension

In this paper, we investigate the optimal control problems governed by elliptic equations with integral constraint for state variable in one dimension by Legendre-Galerkin spectral methods. We deduce optimal conditions of the optimal control problems. Meanwhile, we obtain an a priori error estimate and a posteriori error estimator. Furthermore, we obtain an explicit formula of the a posteriori error estimator by orthogonal properties of Legendre polynomials. Finally, we present numerical examples to confirm our analytical results.