AI 中文总结
研究具有拟最小化约束的非局部最优控制问题，通过新约束选择，在能量最小化器可能不唯一情况下，证明解的存在性，并得到非局部最优控制问题最小化器的收敛结果，相比之前有更强收敛性。
AI 中文摘要
本文研究了一类新的非局部最优控制问题，其约束涉及拟凸能量最小化器的近似。这些问题由分数参数\(s\in(0,1)\)和时间范围参数\(\delta>0\)参数化，能量密度取决于非局部分数梯度。这里，约束包括找到关于拟凸能量的拟最小化器。尽管相关能量可能没有唯一的最小化器，但我们可以证明这类控制问题解的存在性。Cueto - Siktar 2026中的约束是找到能量的全局最小化器，该问题的主要局限是无法证明非局部控制问题的解收敛到相应局部偏微分方程约束最优控制问题的解。而我们新的约束选择得到了更强的收敛结果，即对于依赖非局部梯度的一般成本泛函的非局部最优控制问题，我们得到了最小化器的收敛性。
英文摘要
This paper studies a new class of nonlocal optimal control problems where the constraints involve approximations of minimizers of quasiconvex energies. These problems are parameterized by a fractional parameter $s \in (0, 1)$ and a horizon parameter $δ> 0$, and the energy density depends on a nonlocal fractional gradient. Here, the constraint consists of finding quasi-minimizers with respect to the quasiconvex energy. Despite the fact that the energies of interest may not have unique minimizers, we may prove the existence of solutions to this class of control problems. The constraint in Cueto-Siktar 2026 was finding global minimizers of the energy, and this problem's main limitation was an inability to prove convergence of solutions for the nonlocal control problems to those of a corresponding local, PDE-constrained optimal control problem. While this issue arises from the lack of uniqueness of minimizers for the constraining energy, we get stronger convergence results with our new choice of constraints. Namely, we obtain convergence of minimizers for nonlocal optimal control problems that have a general cost functional depending on the nonlocal gradient.
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