We consider a cantilever beam which possesses a possibly non-uniform permanent magnetization, and whose shape is controlled by an applied magnetic field. We model the beam as a plane elastic curve and we suppose that the magnetic field acts upon the beam by means of a distributed couple that pulls the magnetization towards its direction. Given a list of target shapes, we look for a design of the magnetization profile and for a list of controls such that the shapes assumed by the beam when acted upon by the controls are as close as possible to the targets, in an averaged sense. To this effect, we formulate and solve an optimal design and control problem leading to the minimization of a functional which we study by both direct and indirect methods. In particular, we prove that minimizers exist, solve the associated Lagrange-multiplier formulation (besides non-generic cases), and are unique at least for sufficiently low intensities of the controlling magnetic fields. To achieve the latter result, we use two nested fixed-point arguments relying on the Lagrange-multiplier formulation of the problem, a method which also suggests a numerical scheme. Various relevant open question are also discussed.

中文翻译：

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磁性弹性体的形状编程

我们考虑一个悬臂梁，它可能具有不均匀的永久磁化强度，其形状由外加磁场控制。我们将梁建模为平面弹性曲线，并假设磁场通过将磁化拉向其方向的分布式耦合作用在梁上。给定目标形状列表，我们寻找磁化分布的设计和控件列表，以便在平均意义上，当控件作用时，光束所呈现的形状尽可能接近目标。为此，我们制定并解决了优化设计和控制问题，从而使我们通过直接和间接方法研究的函数最小化。特别是，我们证明了极小值存在，解决相关的拉格朗日乘数公式（除了非一般情况），并且至少对于足够低的控制磁场强度是唯一的。为了实现后一个结果，我们使用两个嵌套的定点参数，依赖于问题的拉格朗日乘数公式，这种方法也提出了一种数值方案。还讨论了各种相关的未解决问题。