Spectral mixed Galerkin method for state constrained optimal control problem governed by the first bi-harmonic equation

An optimal control problem governed by the first bi-harmonic equation with the integral constraint for the state and its spectral approximations based on a mixed formulation are investigated. The optimality conditions of the exact and the discrete optimal control systems are derived. The a priori error estimates of high order spectral accuracy are obtained. Furthermore, a simple and efficient iterative algorithm is proposed to solve mixed discrete system. Some numerical examples are performed to verify the theoretical results.