Shape Programming of a Magnetic Elastica

We consider a cantilever beam which possesses a possibly non-uniform permanent magnetization, $\alpha(s)$, and whose shape $\vartheta(s)$ is controlled by an applied magnetic field, $\vec h$. 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 $(\bar \vartheta_i(s))_{i=1,\ldots n}$ of $n$ target shapes, we look for a design of the magnetization profile $\alpha(s)$ and for a list $(\vec h_i)_{i=1,\ldots n}$ of controls such that the shapes $(\vartheta_i(s))_{i=1,\ldots n}$ 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 uniqueness of minimizers for sufficiently low intensities of the controlling magnetic fields. To this aim, we use two nested fixed-point arguments relying on the Lagrange-multiplier formulation of the problem, a method which also suggests a numerical scheme.