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

Abstract

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 $α\in(0,1)$ 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: $O(τ^{\min({1}/{2}+α-ε,1)}+h^2)$ in the discrete $L^2(0,T;L^2(Ω))$ norm and $O(τ^{α-ε}+\ell_h^2h^2)$ in the discrete $L^\infty(0,T;L^2(Ω))$ norm, with any small $ε>0$ 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.