Galerkin spectral method for a fourth-order optimal control problem with H¹-norm state constraint[Formula presented]

In this article, we are concerned with the Galerkin spectral approximation of an H¹-norm state-constrained optimal control problem governed by a fourth-order partial differential equation (PDE). In the investigations, the optimality conditions for both the original control problem and its spectral approximation problem are, respectively, established, which are helpful to analyze the solution property of the spectral approximation problem. A priori and a posteriori error estimates of the spectral approximation problem are, therewith, built in detail. Finally, two numerical examples are carried out to validate the theoretical analysis.