Spectral approximation for optimal control problems governed by first bi-harmonic equation

We investigate in the article the L-2-norm state constrained control problem with first bi-harmonic equation. First, the optimality conditions of the control problem are derived carefully, and spectral discretization of the problem is established. Based on the property of projection operator, we establish a priori error analysis for control variable, state, and adjoint state variable. Furthermore, the efficient projected gradient algorithm is proposed, and the numerical results we obtained verify the analytical results that it can provide high order accuracy and fast convergence rate.