On some geometric properties of extremals in the optimal retargeting problem for spherically symmetric rigid bodies

We study an optimal control problem for the spatial retargeting of a rotating rigid body with unit tensor of inertia. We find new geometric properties of the extremals in this variational problem in the nondegenerate case. We give a detailed description of the extremal “collapse” effect and its relation to the planar manoeuvre. We also show that each extremal has some rank properties. The results are based on the analysis of the system of equations obtained by applying the formalism of the Pontryagin maximum principle and on the use of first integrals.