分数拉普拉奇的双线性最优控制：分析与离散化

IF 2.8 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis Pub Date : 2024-06-04 DOI:10.1137/23m154947x

Francisco Bersetche, Francisco Fuica, Enrique Otárola, Daniel Quero

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引用次数: 0

摘要

SIAM 数值分析期刊》第 62 卷第 3 期第 1344-1371 页，2024 年 6 月。摘要我们采用分数拉普拉斯算子的积分定义，研究了一个 Lipschitz 域上的最优控制问题，该问题涉及一个作为状态方程的分数椭圆 PDE 和一个作为系数进入状态方程的控制变量；同时还考虑了控制变量的点约束。我们确定了最优解的存在性，并分析了一阶最优条件和必要且充分的二阶最优条件。我们还分析了最优变量的正则性估计。我们开发了两种有限元离散化策略：一种是控制变量不离散化的半离散方案，另一种是控制变量用片断常数函数离散化的完全离散方案。对于这两种方案，我们分析了离散化的收敛特性，并得出了误差估计值。

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Bilinear Optimal Control for the Fractional Laplacian: Analysis and Discretization

SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1344-1371, June 2024.
Abstract. We adopt the integral definition of the fractional Laplace operator and study an optimal control problem on Lipschitz domains that involves a fractional elliptic PDE as the state equation and a control variable that enters the state equation as a coefficient; pointwise constraints on the control variable are considered as well. We establish the existence of optimal solutions and analyze first- and necessary and sufficient second-order optimality conditions. Regularity estimates for optimal variables are also analyzed. We develop two finite element discretization strategies: a semidiscrete scheme in which the control variable is not discretized and a fully discrete scheme in which the control variable is discretized with piecewise constant functions. For both schemes, we analyze the convergence properties of discretizations and derive error estimates.

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来源期刊

SIAM Journal on Numerical Analysis 数学-应用数学

CiteScore

4.80

自引率

6.90%

发文量

110

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Numerical Analysis (SINUM) contains research articles on the development and analysis of numerical methods. Topics include the rigorous study of convergence of algorithms, their accuracy, their stability, and their computational complexity. Also included are results in mathematical analysis that contribute to algorithm analysis, and computational results that demonstrate algorithm behavior and applicability.