Equivalent a Posteriori Error Estimator of Spectral Approximation for Control Problems with Integral Control-State Constraints in One Dimension

In this paper, we investigate the Galerkin spectral approximation for ellipticcontrol problems with integral control and state constraints. Firstly, an a posteriori errorestimator is established, which can be acted as the equivalent indicator with explicitexpression. Secondly, appropriate base functions of the discrete spaces make it probableto solve the discrete system. Numerical test indicates the reliability and efficiencyof the estimator, and shows the proposed method is competitive for this class of controlproblems. These discussions can certainly be extended to two- and three-dimensionalcases.