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A Fast Gradient Projection Method for a Constrained Fractional Optimal Control

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Abstract

Optimal control problems governed by a fractional diffusion equation tends to provide a better description than one by a classical second-order Fickian diffusion equation in the context of transport or conduction processes in heterogeneous media. However, the fractional control problem introduces significantly increased computational complexity and storage requirement than the corresponding classical control problem, due to the nonlocal nature of fractional differential operators. We develop a fast gradient projection method for a pointwise constrained optimal control problem governed by a time-dependent space-fractional diffusion equation, which requires the computational cost from \(O(M N^3)\) of a conventional solver to \(O(M N\log N)\) and memory requirement from \(O(N^2)\) to O(N) for a problem of size N and of M time steps. Numerical experiments show the utility of the method.

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Author information

Authors and Affiliations

  1. School of Mathematics, Shandong Univeristy, Jinan, Shandong, China

    Ning Du

  2. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Hong Wang

  3. KBS, University of Kent, Canterbury, CT2 7NF, UK

    Wenbin Liu

Authors
  1. Ning Du
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  2. Hong Wang
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  3. Wenbin Liu
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Corresponding author

Correspondence to Hong Wang.

Additional information

This work was supported by the National Natural Science Foundation of China under Grants 11371229, 91130010, and 11471194, by the National Science Foundation under Grants EAR-0934747 and DMS-1216923 and, by the China Scholarship Council (File No. 201308370102).

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Du, N., Wang, H. & Liu, W. A Fast Gradient Projection Method for a Constrained Fractional Optimal Control. J Sci Comput 68, 1–20 (2016). https://doi.org/10.1007/s10915-015-0125-1

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  • DOI: https://doi.org/10.1007/s10915-015-0125-1

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