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We investigate the Galerkin spectral approximation of an optimal control problem governed by a fourth-order partial differential equation (PDE), in which an-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-error estimate for the state and co-state but also-error estimate for the control, state and co-state. Finally, three numerical examples are executed to validate the theoretical analysis.

Galerkin spectral approximation for optimal control problem of a fourth-order equation with L 2 -norm control constraint