<p>This paper addresses an inverse problem concerning the identification of an unknown initial value in a one-dimensional fractional diffusion equation. The initial value is reconstructed from the final noisy data. To address this inverse problem, we first examine the well-posedness of the direct problem. Then, the inverse problem is reformulated as a regularized optimal control problem by employing the least squares method. After that, we demonstrate the existence and stability of solutions to the optimal control problem. In the meanwhile, we prove the Fréchet differentiability of the cost function and establish its convexity. Furthermore, we reconstruct the initial value via using the conjugate gradient method. Finally, to illustrate the effectiveness of the proposed method, we present results from a series of numerical experiments.</p>