Shape programming of a magnetic elastica

Technological advancements have made it possible to develop devices capable of changing their shape in response to external stimuli, with promising applications in microsurgery or targeted drug delivery, among other fields. Rather than solely relying on human intuition to design such devices, shape programming aims to systematically design the device’s morphology and external stimuli to achieve a desired shape.In this work, we focus on a cantilever beam with permanent magnetization of constant intensity but variable direction and model it as a planar elastica. The shape of the beam can be controlled by a spatially constant external magnetic field. Given a set of target shapes, we aim to determine the magnetization for the beam, along with a list of external magnetic fields, such that the shapes attained by the beam, when subjected to these fields,closely approximate the target shapes.For this purpose, we formulate an optimal design-control problem aiming to minimize the deviation from the attained shapes to the target shapes. We prove the existence of a minimizer via the direct method and derive necessary optimality conditions using the associated Lagrange multiplier formulation. For sufficiently low magnetic field intensities,we establish uniqueness of the minimizer through nested fixed-point techniques based on the optimality conditions. In addition, we implement a gradient descent algorithm and present numerical results demonstrating its effectiveness. Lastly, we address control constraints and formulate a reduced optimal design-control problem using the control-to-state operator followed by exploring the existence of a minimizer and necessary optimality conditions.