Shape programming of a magnetic elastica

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