Galerkin spectral approximations for optimal control problems governed by the fourth order equation with an integral constraint on state

An integral constraint on state optimal control problem governed by the first bi-harmonic equation is analyzed and its Legendre Galerkin spectral approximations are constructed. The optimality conditions of the continuous and the discretized optimal control problems are presented by the Kuhn-Tucker condition, respectively. The a priori error estimates of high accuracy are obtained. Furthermore, an efficient projection algorithm is proposed, and the convergent rate is discussed with precise details. Ample numerical examples are carried out to validate the theoretical results.