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

TL;DR

This paper develops and analyzes a numerical scheme for an optimal control problem constrained by a subdiffusion equation with fractional time derivatives, providing error estimates and supporting numerical experiments.

Contribution

It introduces a fully discrete numerical scheme with proven convergence rates for a subdiffusion-constrained optimal control problem, utilizing maximal regularity techniques.

Findings

Convergence order of $O( au^{ ext{min}(1/2+ ext{alpha}- ext{epsilon},1)}+h^2)$ in $L^2$ norm.

Convergence order of $O( au^{ ext{alpha}- ext{epsilon}}+ ext{log}(2+1/h)^2 h^2)$ in $L^ Infty$ norm.

Numerical experiments confirm the theoretical error estimates.

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 $O(\tau^{\alpha-\epsilon}+\ell_h^2h^2)$ 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.