SIAM Journal on Control and Optimization, Volume 59, Issue 1, Page 393-416, January 2021.
This paper addresses, from both theoretical and numerical standpoints, the problem of optimal control of hyperelastic materials characterized by means of polyconvex stored energy functionals. Specifically, inspired by Günnel and Herzog [Front. Appl. Math. Stat., 2 (2016)], a bio-inspired type of external action or control, which resembles the electro-activation mechanism of the human heart, is considered in this paper. The main contribution resides in the consideration of tracking-type cost functionals alternative to those generally used in this field, where the $L^2$ norm of the distance to a given target displacement field is the preferred option. Alternatively, the Hausdorff metric is, for the first time, explored in the context of optimal control in hyperelasticity. The existence of a solution for a regularized version of the optimal control problem is proved. A gradient-based method, which makes use of the concept of shape derivative, is proposed as a numerical resolution method. A series of numerical examples are included illustrating the viability and applicability of the Hausdorff metric in this new context. Furthermore, although not pursued in this paper, it must be emphasized that in contrast to $L^2$ norm tracking-cost functional types, the Hausdorff metric permits the use of potentially very different computational domains for both the target and the actuated soft continuum.

中文翻译：

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利用Hausdorff距离函数的软材料最优控制。

SIAM控制与优化杂志，第59卷，第1期，第393-416页，2021年1月。
本文从理论和数值两个角度解决了以多凸储能功能为特征的超弹性材料的最优控制问题。具体而言，受到Günnel和Herzog [Front。应用 数学。（Stat。，2（2016）），一种类似于人类心脏电激活机制的生物启发型外部作用或控制方法。主要的贡献在于考虑了跟踪型成本函数，该函数可替代该领域中常用的跟踪函数，在该函数中，到给定目标位移场的距离的$ L ^ 2 $范数是首选。或者，首次在超弹性的最佳控制范围内探索Hausdorff度量。证明了最优控制问题的正规化解的存在。提出了一种利用形状导数概念的基于梯度的方法作为数值解析方法。包括一系列数值示例，它们说明了在这种新情况下Hausdorff度量的可行性和适用性。此外，尽管本文未进行探讨，但必须强调的是，与$ L ^ 2 $范数跟踪成本函数类型相反，Hausdorff度量允许对目标和激活的软连续体使用可能非常不同的计算域。包括一系列数值示例，它们说明了在这种新情况下Hausdorff度量的可行性和适用性。此外，尽管本文未进行探讨，但必须强调的是，与$ L ^ 2 $范数跟踪成本函数类型相反，Hausdorff度量允许对目标和激活的软连续体使用可能非常不同的计算域。包括一系列数值示例，它们说明了在这种新情况下Hausdorff度量的可行性和适用性。此外，尽管本文未进行探讨，但必须强调的是，与$ L ^ 2 $范数跟踪成本函数类型相反，Hausdorff度量允许对目标和激活的软连续体使用可能非常不同的计算域。