Error Analysis of Fully Discrete Finite Element Approximations to an Optimal Control Problem Governed by a Time-Fractional PDE

The optimality system is derived for an optimal control problem governed by a time-fractional PDE. A fully discrete finite element method along with convolution quadrature for time discretization are proposed to determine approximate solutions of the optimality system. Almost optimal convergence of $O(\tau|\ln\tau|+h^2)$ is proved for the proposed numerical method, relying only on regularity assumptions on the data, without extra assumptions on the regularity of solutions of the optimality system.