ANALYSIS AND DISCRETIZATION FOR AN OPTIMAL CONTROL PROBLEM OF A VARIABLE-COEFFICIENT RIESZ-FRACTIONAL DIFFUSION EQUATION WITH POINTWISE CONTROL CONSTRAINTS*

doi:10.1007/s10473-023-0210-8

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (2): 640-654.doi: 10.1007/s10473-023-0210-8

ANALYSIS AND DISCRETIZATION FOR AN OPTIMAL CONTROL PROBLEM OF A VARIABLE-COEFFICIENT RIESZ-FRACTIONAL DIFFUSION EQUATION WITH POINTWISE CONTROL CONSTRAINTS^*

Zhaojie Zhou¹, Fangyuan Wang¹, Xiangcheng Zheng^2,†

1. School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China;
2. School of Mathematics, Shandong University, Jinan 250100, China

收稿日期:2021-10-29 修回日期:2022-02-08 出版日期:2023-03-25 发布日期:2023-04-12
通讯作者: †Xiangcheng Zheng, E-mail: zhengxch@outlook.com.
作者简介:Zhaojie Zhou, E-mail: zhouzhaojie@sdnu.edu.cn; Fangyuan Wang, E-mail: wfy_sdnu@163.com
基金资助:
The first author was supported by the National Natural Science Foundation of China (11971276, 12171287), and by the Natural Science Foundation of Shandong Province (ZR2016JL004). The third author was supported by the China Postdoctoral Science Foundation (2021TQ0017, 2021M700244), and by the International Postdoctoral Exchange Fellowship Program (Talent-Introduction Program) (YJ20210019).

ANALYSIS AND DISCRETIZATION FOR AN OPTIMAL CONTROL PROBLEM OF A VARIABLE-COEFFICIENT RIESZ-FRACTIONAL DIFFUSION EQUATION WITH POINTWISE CONTROL CONSTRAINTS^*

Zhaojie Zhou¹, Fangyuan Wang¹, Xiangcheng Zheng^2,†

1. School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China;
2. School of Mathematics, Shandong University, Jinan 250100, China

Received:2021-10-29 Revised:2022-02-08 Online:2023-03-25 Published:2023-04-12
Contact: †Xiangcheng Zheng, E-mail: zhengxch@outlook.com.
About author:Zhaojie Zhou, E-mail: zhouzhaojie@sdnu.edu.cn; Fangyuan Wang, E-mail: wfy_sdnu@163.com
Supported by:
The first author was supported by the National Natural Science Foundation of China (11971276, 12171287), and by the Natural Science Foundation of Shandong Province (ZR2016JL004). The third author was supported by the China Postdoctoral Science Foundation (2021TQ0017, 2021M700244), and by the International Postdoctoral Exchange Fellowship Program (Talent-Introduction Program) (YJ20210019).

摘要/Abstract

摘要： We present a mathematical and numerical study for a pointwise optimal control problem governed by a variable-coefficient Riesz-fractional diffusion equation. Due to the impact of the variable diffusivity coefficient, existing regularity results for their constant-coefficient counterparts do not apply, while the bilinear forms of the state (adjoint) equation may lose the coercivity that is critical in error estimates of the finite element method. We reformulate the state equation as an equivalent constant-coefficient fractional diffusion equation with the addition of a variable-coefficient low-order fractional advection term. First order optimality conditions are accordingly derived and the smoothing properties of the solutions are analyzed by, e.g., interpolation estimates. The weak coercivity of the resulting bilinear forms are proven via the Garding inequality, based on which we prove the optimal-order convergence estimates of the finite element method for the (adjoint) state variable and the control variable. Numerical experiments substantiate the theoretical predictions.

关键词: Riesz-fractional diffusion equation, variable coefficient, optimal control, finite element method, Garding inequality, optimal-order error estimate

Abstract: We present a mathematical and numerical study for a pointwise optimal control problem governed by a variable-coefficient Riesz-fractional diffusion equation. Due to the impact of the variable diffusivity coefficient, existing regularity results for their constant-coefficient counterparts do not apply, while the bilinear forms of the state (adjoint) equation may lose the coercivity that is critical in error estimates of the finite element method. We reformulate the state equation as an equivalent constant-coefficient fractional diffusion equation with the addition of a variable-coefficient low-order fractional advection term. First order optimality conditions are accordingly derived and the smoothing properties of the solutions are analyzed by, e.g., interpolation estimates. The weak coercivity of the resulting bilinear forms are proven via the Garding inequality, based on which we prove the optimal-order convergence estimates of the finite element method for the (adjoint) state variable and the control variable. Numerical experiments substantiate the theoretical predictions.

Key words: Riesz-fractional diffusion equation, variable coefficient, optimal control, finite element method, Garding inequality, optimal-order error estimate

Zhaojie Zhou, Fangyuan Wang, Xiangcheng Zheng. ANALYSIS AND DISCRETIZATION FOR AN OPTIMAL CONTROL PROBLEM OF A VARIABLE-COEFFICIENT RIESZ-FRACTIONAL DIFFUSION EQUATION WITH POINTWISE CONTROL CONSTRAINTS^*[J]. 数学物理学报(英文版), 2023, 43(2): 640-654.

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ANALYSIS AND DISCRETIZATION FOR AN OPTIMAL CONTROL PROBLEM OF A VARIABLE-COEFFICIENT RIESZ-FRACTIONAL DIFFUSION EQUATION WITH POINTWISE CONTROL CONSTRAINTS^*

ANALYSIS AND DISCRETIZATION FOR AN OPTIMAL CONTROL PROBLEM OF A VARIABLE-COEFFICIENT RIESZ-FRACTIONAL DIFFUSION EQUATION WITH POINTWISE CONTROL CONSTRAINTS^*

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