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<p>In this paper, an optimal control problem governed by a time-fractional diffusion equation is meticulously approximated based on Crank-Nicolson discretization in time to achieve higher temporal convergence order. Under absent control constraints, the regularity results on the second-order time derivatives of the control, state and adjoint variables in the optimality system are estimated. Together with the linear finite element discretization in space, we derive the optimality conditions of the discretized optimal control system and rigorously analyze the temporal error estimates of the control, state and adjoint variables only concerning the regularity property of the given data. The theoretical result indicates that our proposed Crank-Nicolson discretization scheme for the considered fractional optimal control problem converges by the optimal order of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3051_Article_IEq1.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="102" /> </InlineMediaObject> <EquationSource Format="TEX">\(O(\tau ^{\min \{\frac{3}{2}+\alpha ,2\}})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>τ</mi> <mrow> <mo movablelimits="true">min</mo> <mo stretchy="false">{</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> <mo>+</mo> <mi>α</mi> <mo>,</mo> <mn>2</mn> <mo stretchy="false">}</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in time, which is verified in numerical examples.</p>

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Error Analysis of Crank-Nicolson Scheme for an Optimal Control Problem with Time-Fractional Diffusion Equation

  • 期刊论文

  • 发表日期：

    2025年9月9日

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  • Xiaoyu Chen,
  • Wenyi Tian

摘要

In this paper, an optimal control problem governed by a time-fractional diffusion equation is meticulously approximated based on Crank-Nicolson discretization in time to achieve higher temporal convergence order. Under absent control constraints, the regularity results on the second-order time derivatives of the control, state and adjoint variables in the optimality system are estimated. Together with the linear finite element discretization in space, we derive the optimality conditions of the discretized optimal control system and rigorously analyze the temporal error estimates of the control, state and adjoint variables only concerning the regularity property of the given data. The theoretical result indicates that our proposed Crank-Nicolson discretization scheme for the considered fractional optimal control problem converges by the optimal order of \(O(\tau ^{\min \{\frac{3}{2}+\alpha ,2\}})\) O ( τ min { 3 2 + α , 2 } ) in time, which is verified in numerical examples.