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Optimal Control of Fractional Elliptic PDEs with State Constraints and Characterization of the Dual of Fractional-Order Sobolev Spaces
Journal of Optimization Theory and Applications ( IF 1.9 ) Pub Date : 2020-05-18 , DOI: 10.1007/s10957-020-01684-z
Harbir Antil , Deepanshu Verma , Mahamadi Warma

This paper introduces the notion of state constraints for optimal control problems governed by fractional elliptic PDEs of order $s \in (0,1)$. There are several mathematical tools that are developed during the process to study this problem, for instance, the characterization of the dual of the fractional order Sobolev spaces and well-posedness of fractional PDEs with measure-valued datum. These tools are widely applicable. We show well-posedness of the optimal control problem and derive the first order optimality conditions. Notice that the adjoint equation is a fractional PDE with measure as the right-hand-side datum. We use the characterization of the fractional order dual spaces to study the regularity of the state and adjoint equations. We emphasize that the classical case ($s=1$) was considered by E. Casas in \cite{ECasas_1986a} but almost none of the existing results are applicable to our fractional case.

中文翻译:

带状态约束的分数椭圆偏微分方程的最优控制和分数阶 Sobolev 空间对偶的表征

本文介绍了由 $s \in (0,1)$ 阶分数椭圆 PDE 控制的最优控制问题的状态约束概念。在研究这个问题的过程中开发了几种数学工具,例如,分数阶 Sobolev 空间的对偶特征和具有测量值数据的分数阶偏微分方程的适定性。这些工具是广泛适用的。我们展示了最优控制问题的适定性并推导出一阶最优条件。请注意,伴随方程是一个分数偏微分方程,其右侧数据为测度。我们使用分数阶对偶空间的表征来研究状态和伴随方程的正则性。我们强调 E 考虑了经典情况 ($s=1$)。
更新日期:2020-05-18
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