Clebsch optimal control formulation in mechanics

Pages: 41 - 79, Volume 3, Issue 1, March 2011

doi:10.3934/jgm.2011.3.41       Abstract        References        Full Text (597.2K)       Related Articles

François Gay-Balmaz - Centre National de la Recherche Scientifique (CNRS), Laboratoire de Météorologie Dynamique, École Normale Supérieure, Paris, France (email)
Tudor S. Ratiu - Section de Mathématiques and Bernoulli Center, Ecole Polytechnique Fédérale de Lausanne, Lausanne, CH-1015, Switzerland (email)

Abstract: This paper introduces and studies a class of optimal control problems based on the Clebsch approach to Euler-Poincaré dynamics. This approach unifies 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 flow using the back-to-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.

Keywords:  Optimal control, Clebsch variables, momentum map, Euler-Poincaré equation, double bracket equation, Euler equation, normal metric, geodesic.
Mathematics Subject Classification:  49J15, 49J20, 37K05, 58E30, 70H30.

Received: September 2010;      Revised: April 2011;      Published: April 2011.

 References