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

Jin, B; Li, B; Zhou, Z; (2018) Pointwise-in-time error estimates for an optimal control problem with subdiffusion constraint. IMA Journal of Numerical Analysis , 40 (1) pp. 377-404. 10.1093/imanum/dry064. Green open access

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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 ∈ (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 t, we establish the following order of convergence for the numerical solutions of the optimal control problem O(t^{(min(1/2+α-ε, 1)} +h²) in the discrete L²(0;T;L²(Ω)) norm and O(t^{α-ε} + l^{2/h}h²) in the discrete L∞(0;T;L²(Ω)) norm, with any small ε > 0 and l_{h} = 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.

Type: Article
Title: Pointwise-in-time error estimates for an optimal control problem with subdiffusion constraint
Open access status: An open access version is available from UCL Discovery
DOI: 10.1093/imanum/dry064
Publisher version: https://doi.org/10.1093/imanum/dry064
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: optimal control, time-fractional diffusion, L1 scheme, convolution quadrature, pointwise-in-time error estimate, maximal regularity
UCL classification: UCL
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UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Computer Science
URI: https://discovery.ucl.ac.uk/id/eprint/10056720
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