Optimal control on Riemannian manifolds with potential fields

This paper is an extension of the work done by I.I. Hussein and A.M. Bloch (2004), where necessary conditions for minimizing the cost function for a trajectory that evolves on a Riemannian manifold and satisfies a second order differential equation together with some interpolation, smoothness and motion constraints were derived. In this paper, we investigate the inclusion of a generalized potential field in the dynamics. This problem is motivated by two-spacecraft interferometric imaging applications, where the formation is evolving in some generalized potential field. The corresponding necessary conditions are derived and connections are made with previous results. The paper is concluded with an example.