A computational approach based on the Legendre-Galerkin method for solving a distributed optimal control problem constrained by the biharmonic equation

This paper presents a Legendre-Galerkin spectral method to compute the solution of a distributed optimal control problem (OCP) constrained by the biharmonic equation on regular and irregular domains. First, the optimality system is obtained by the Karush–Kuhn–Tucker optimality conditions. Next, it is discretized by the proposed method, and a coupled matrix equation is obtained. Finally, the arising system is solved by using the Kronecker product. Using the appropriate base functions causes a system with sparse matrices. Also, one of the advantages of this approach is its high accuracy for solving fourth-order problems. The error estimate is also investigated theoretically. Many numerical examples are included to demonstrate the accuracy and efficiency of the proposed approach.