Clebsch optimal control formulation in mechanics

This paper introduces and studies a class of optimal control problems based on the Clebsch approach to Euler-Poincaré dynamics. This approach unifles and generalizes a wide range of examples appearing in the literature: the symmetric formulation of N-dimensional rigid body and its generalization to other matrix groups; optimal control for ideal ow using the backto- labels map; the double bracket equations associated to symmetric spaces. New examples are provided such as the optimal control formulation for the N-Camassa-Holm equation and a new geodesic interpretation of its singular solutions.