Legendre pseudo-spectral method for optimal control problem governed by a time-fractional diffusion equation

This paper presents a numerical scheme for optimal control problem governed by a time-fractional diffusion equation based on a Legendre pseudo-spectral method for space discretization and a finite difference method for time discretization. Lagrange interpolating basis polynomials are used to approximate the state, and the differentiation matrix is derived to discrete the spatial derivative. We also discuss the fully discrete scheme for the control problem. A finite difference method developed in Lin and Xu [Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys. 225 (2007), pp. 1533-1552] is used to discretize the time-fractional derivative. A fully discrete first-order optimality condition is developed based on the first discretize, then optimize' approach. Furthermore, we design the projected gradient algorithm based on the fully discrete optimality conditions. Numerical examples are given to illustrate the feasibility of the proposed method.