SIAM Journal on Control and Optimization, Volume 58, Issue 4, Page 1769-1794, January 2020.
We consider a problem of optimal distribution of conductivities in a system governed by a nonlocal diffusion law. The problem stems from applications in optimal design and more specifically topology optimization. We propose a novel parametrization of nonlocal material properties. With this parametrization the nonlocal diffusion law in the limit of vanishing nonlocal interaction horizons converges to the famous and ubiquitously used generalized Laplacian with SIMP (solid isotropic material with penalization) material model. The optimal control problem for the limiting local model is typically ill-posed and does not attain its infimum without additional regularization. Surprisingly, its nonlocal counterpart attains its global minima in many practical situations, as we demonstrate in this work. In spite of this qualitatively different behavior, we are able to partially characterize the relationship between the nonlocal and the local optimal control problems. We also complement our theoretical findings with numerical examples, which illustrate the viability of our approach to optimal design practitioners.

中文翻译：

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传导系数的非局部控制：良好定理和收敛到局部极限

SIAM控制与优化杂志，第58卷，第4期，第1769-1794页，2020年1月。
我们考虑由非局部扩散定律控制的系统中电导率的最佳分布问题。问题源于优化设计中的应用，更具体地讲是拓扑优化中的应用。我们提出了一种非局部材料特性的新型参数化方法。通过这种参数化，在非局部交互作用范围消失的极限内的非局部扩散定律收敛于著名的和普遍使用的带有SIMP（带有罚分的固体各向同性材料）的广义拉普拉斯算子。限制局部模型的最优控制问题通常是不适定的，如果不进行其他正则化处理就无法获得最优控制问题。令人惊讶的是，正如我们在这项工作中所展示的，其非本地对手在许多实际情况下都达到了全球最低要求。尽管这种行为在质量上有所不同，我们能够部分刻画非局部和局部最优控制问题之间的关系。我们还通过数字示例补充了我们的理论发现，这些示例说明了我们对最佳设计从业人员的可行性。