  In this paper, we consider a ferromagnetic material of ellipsoidal shape. The
associated magnetic moment then has two asymptotically stable opposite
equilibria, of the form $\pm\overline{m}$. In order to use these materials for
memory storage purposes, it is necessary to know how to control the magnetic
moment. We use as a control variable a spatially uniform external magnetic
field and consider the question of flipping the magnetic moment, i.e., changing
it from the $+\overline{m}$ configuration to the $-\overline{m}$ one, in
minimal time. Of course, it is necessary to impose restrictions on the external
magnetic field used. We therefore include a constraint on the $L^\infty$ norm
of the controls, assumed to be less than a threshold value $U$. We show that,
generically with respect to the dimensions of the ellipsoid, there is a minimal
value of $U$ for this problem to have a solution. We then characterize it
precisely. Finally, we investigate some particular configurations associated to
geometries enjoying symmetries properties and show that in this case the
magnetic moment can be controlled in minimal time without imposing a threshold
condition on $U$.
