A Spectral Method for Optimal Control Problems Governed by the Time Fractional Diffusion Equation with Control Constraints

In this paper, we study the fractional optimal control problem and its spectral approximation. The problem under investigation consists in finding the optimal solution governed by the time fractional diffusion equation with constraints on the control variable. We construct a suitable weak formulation, study its well-posedness, and design a Galerkin spectral method for its numerical solution. The main contribution of the paper includes: (1) a priori error estimates for the space-time spectral approximation is derived; (2) a projection gradient algorithm is designed to efficiently solve the discrete minimization problem; (3) some numerical experiments are carried out to confirm the efficiency of the proposed method. The obtained numerical results show that the convergence is exponential for smooth exact solutions.