Elastic curves as solutions of Riemannian and sub-Riemannian control problems

We consider the nonlinear dynamic interpolation problem on Riemannian manif olds and, in particular, on connected and compact Lie groups. Basically we force the dynamic variables of a control system to pass through specific po ints in the configuration space, while minimizing a certain energy function , by a suitable choice of the controls. The energy function we consider dep ends on the velocity and acceleration along trajectories. The solution curv es can be seen as generalizations of the classical splines in tension for t he Euclidean space. The relations with sub-Riemannian optimal control probl ems are explained.