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

Abstract

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].Acta mathematica scientia,Series B, 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^*

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