Error Analysis of Spectral Method for Fractional Optimal Control Problem with Non-Homogeneous Boundary Condition

In this paper, we investigate an optimal control problem governed by a time fractional diffusion equation with non-homogeneous boundary conditions. We construct a suitable weak formulation, study its well-posedness, and develop a Galerkin spectral method for its numerical solution. The main contributions of the paper are: 1) the study of optimality conditions, which is crucial for analyzing the optimal control problem; 2) the establishment of a priori error estimates for the space-time spectral approximation; 3) the derivation of a posteriori error estimates for the state, costate, and control approximations; 4) numerical experiments that confirm the efficiency of the proposed method. Numerical tests are carried out to confirm the theoretical results.