A Legendre Galerkin Spectral Method For Optimal Control Problems

Spectral method is a kind of both old and new numerical methods for solving partial differential equations. It is proved to be an efficient numerical method by a lot of practical computation and is widely applied to a lot of areas on scientific and engineering.In this paper, we have researched on a Legendre Galerkin method for elliptic optimal control problems. Firstly, A Legendre Galerkin spectral approximation is introduced and the error estimates, stability and convergence are obtained. By choosing the appropriate basis functions, the stiff matrix of the discretization equations is sparse. And the efficiency of the algorithm is improved by using fast Legendre transform. Two numerical experiments of the one dimensional and two dimensional respectively are presented to confirm the error estimates. The results show the well efficiency.This work is among the first to apply the spectral method to compute the distributed optimal control problems. It paved a way for much more work on this area.