SIAM Journal on Control and Optimization, Volume 62, Issue 1, Page 487-508, February 2024.
Abstract. We explore the dual approach to nonlocal optimal control in the coefficients, specifically for a classical min-max problem which in this study is associated with a nonlocal scalar diffusion equation. We reformulate the optimal control problem utilizing a dual variational principle, which is expressed in terms of nonlocal two-point fluxes. We introduce the proper functional space framework to deal with this formulation and establish its well-posedness. The key ingredient is the inf-sup (Ladyzhenskaya–Babuška–Brezzi) condition, which holds uniformly with respect to small nonlocal horizons. As a by-product of this fact, we are able to prove convergence of nonlocal optimal control problems toward their local counterparts in a straightforward fashion.

中文翻译：

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非局部开尔文原理和传导系数非局部控制的双重方法

SIAM 控制与优化杂志，第 62 卷，第 1 期，第 487-508 页，2024 年 2 月。
摘要。我们探索系数中非局部最优控制的对偶方法，特别是对于本研究中与非局部标量扩散方程相关的经典最小-最大问题。我们利用对偶变分原理重新表述最优控制问题，该原理用非局部两点通量表示。我们引入适当的功能空间框架来处理这个公式并建立其适定性。关键要素是 inf-sup (Ladyzhenskaya-Babuška-Brezzi) 条件，该条件对于小的非局部视界一致成立。作为这一事实的副产品，我们能够以直接的方式证明非局部最优控制问题与其局部对应问题的收敛性。