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

作　　者：周兆杰 王方圆 郑祥成 Zhaojie ZHOU;Fangyuan WANG;Xiangcheng ZHENG(School of Mathematics and Statistics,Shandong Normal University,Jinan 250014,China;School of Mathematics,Shandong University,Jinan 250100,China)

机构地区：[1]School of Mathematics and Statistics,Shandong Normal University,Jinan 250014,China [2]School of Mathematics,Shandong University,Jinan 250100,China

出　　处：《Acta Mathematica Scientia》2023年第2期640-654,共15页数学物理学报（B辑英文版）

基　　金：supported by the National Natural Science Foundation of China(11971276,12171287);Natural Science Foundation of Shandong Province(ZR2016JL004);supported by the China Postdoctoral Science Foundation(2021TQ0017,2021M700244);International Postdoctoral Exchange Fellowship Program(Talent-Introduction Program)(YJ20210019)。

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 constantcoefficient 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 

分 类 号：O232[理学—运筹学与控制论]