In this work, we present numerical analysis for a distributed optimal control
problem, with box constraint on the control, governed by a subdiffusion
equation which involves a fractional derivative of order α∈(0,1) in
time. The fully discrete scheme is obtained by applying the conforming linear
Galerkin finite element method in space, L1 scheme/backward Euler convolution
quadrature in time, and the control variable by a variational type
discretization. With a space mesh size h and time stepsize τ, we
establish the following order of convergence for the numerical solutions of the
optimal control problem: O(τmin(1/2+α−ϵ,1)+h2) in the
discrete L2(0,T;L2(Ω)) norm and
O(τα−ϵ+ℓh2h2) in the discrete
L∞(0,T;L2(Ω)) norm, with any small ϵ>0 and
ℓh=ln(2+1/h). The analysis relies essentially on the maximal
Lp-regularity and its discrete analogue for the subdiffusion problem.
Numerical experiments are provided to support the theoretical results.Comment: 20 pages, 6 figure