Geometry and the Dynamic Interpolation Problem

In this paper we consider the dynamic interpolation problem for control systems in which certain dynamic variables of state trajectories are forced to pass through specific points by suitable choices of controls. This problem can be viewed as an extension of the spline problem. Following Noakes, Heinzinger and Paden [16], we give a derivation of suitable interpolating cubic splines on a Riemannian manifold extending the variational approach in Milnor [15]. For the special case of compact Lie groups, the relation with optimal control problems and singular Riemannian Geometry is spelled out in detail.