<p>In this paper, an optimal control problem governed by the forward fractional Feynman-Kac equation is considered, which describes functional distributions of anomalous diffusion and encounters significant challenges arise from the time-space coupled nonlocal operator and its non-commutativity with the Laplacian. First, we investigate the well-posedness of the continuous optimal control problem, derive the first-order optimality conditions and establish the regularity estimates of the solution. Then, the Riemann-Liouville fractional substantial derivative in the equation is approximated by using the backward Euler convolution quadrature formula, and a temporal semi-discrete scheme is proposed for the optimal control problem. Moreover, we rigorously analyze the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\ell ^2(L^2(\Omega ))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>ℓ</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\ell ^{\infty }(L^2(\Omega ))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>ℓ</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> error estimates of the proposed semi-discrete scheme, which exhibits almost optimal convergence of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$O(\tau |\ln \tau |)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">|</mo> <mo>ln</mo> <mi>τ</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, relying only on the regularity assumptions on the data and without extra assumptions on the solution of the optimality system. Finally, we perform the numerical experiments by using the inexact alternating direction method of multipliers (ADMM) algorithm and the piecewise linear finite element method. The numerical results demonstrate the validity of our numerical scheme and verify the theoretical convergence order.</p>

Numerical discretization and error analysis for an optimal control problem governed by forward fractional Feynman-Kac equation

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发表日期：

2025年11月3日

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Liyao Hao,
Wenyi Tian

摘要

In this paper, an optimal control problem governed by the forward fractional Feynman-Kac equation is considered, which describes functional distributions of anomalous diffusion and encounters significant challenges arise from the time-space coupled nonlocal operator and its non-commutativity with the Laplacian. First, we investigate the well-posedness of the continuous optimal control problem, derive the first-order optimality conditions and establish the regularity estimates of the solution. Then, the Riemann-Liouville fractional substantial derivative in the equation is approximated by using the backward Euler convolution quadrature formula, and a temporal semi-discrete scheme is proposed for the optimal control problem. Moreover, we rigorously analyze the $\ell ^2(L^2(\Omega ))$ $ℓ^{2} (L^{2} (Ω))$ and $\ell ^{\infty }(L^2(\Omega ))$ $ℓ^{\infty} (L^{2} (Ω))$ error estimates of the proposed semi-discrete scheme, which exhibits almost optimal convergence of $O(\tau |\ln \tau |)$ $O (τ | ln τ |)$ , relying only on the regularity assumptions on the data and without extra assumptions on the solution of the optimality system. Finally, we perform the numerical experiments by using the inexact alternating direction method of multipliers (ADMM) algorithm and the piecewise linear finite element method. The numerical results demonstrate the validity of our numerical scheme and verify the theoretical convergence order.