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In this work, we present numerical analysis for a distributed optimal controlproblem, with box constraint on the control, governed by a subdiffusionequation which involves a fractional derivative of order $\alpha\in(0,1)$ intime. The fully discrete scheme is obtained by applying the conforming linearGalerkin finite element method in space, L1 scheme/backward Euler convolutionquadrature in time, and the control variable by a variational typediscretization. With a space mesh size $h$ and time stepsize $\tau$, weestablish the following order of convergence for the numerical solutions of theoptimal control problem: $O(\tau^{\min({1}/{2}+\alpha-\epsilon,1)}+h^2)$ in thediscrete $L^2(0,T;L^2(\Omega))$ norm and$O(\tau^{\alpha-\epsilon}+\ell_h^2h^2)$ in the discrete$L^\infty(0,T;L^2(\Omega))$ norm, with any small $\epsilon>0$ and$\ell_h=\ln(2+1/h)$. The analysis relies essentially on the maximal$L^p$-regularity and its discrete analogue for the subdiffusion problem.Numerical experiments are provided to support the theoretical results.

Pointwise-in-time error estimates for an optimal control problem with subdiffusion constraint