Galerkin spectral approximation for optimal control problem of a fourth-order equation with L²-norm control constraint

We investigate the Galerkin spectral approximation of an optimal control problem governed by a fourth-order partial differential equation (PDE), in which an (Formula presented.) -norm constraint on control variable is equipped. First, the optimality conditions for both the original control problem and its spectral approximation problem are, respectively, obtained. Then, a priori error estimates of the spectral approximation problem are established in detail. Next, a posteriori error estimates for the approximation problem are also investigated, which include not only (Formula presented.) -error estimate for the state and co-state but also (Formula presented.) -error estimate for the control, state and co-state. Finally, three numerical examples are executed to validate the theoretical analysis.