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 $\alpha \in (0,1)$ 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 $\tau$, we establish the following order of convergence for the numerical solutions of the optimal control problem: $O({\tau }^{\min (1/2+\alpha -\epsilon ,1)}+h^2)$ in the discrete $L^2(0,T;L^2(\Omega ))$ norm and $O({\tau }^{\alpha -\epsilon }+{\ell }_h^2{h}^2)$ 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.

arXiv comment20 pages, 6 figures

References in corpus (1)

• Numerical analysis of nonlinear subdiffusion equations