Clebsch optimal control formulation in mechanics

François Gay-Balmaz; Tudor S. Ratiu

doi:10.3934/jgm.2011.3.41

Article Contents

2011, Volume 3, Issue 1: 41-79. Doi: 10.3934/jgm.2011.3.41

This issue Previous Article Reduction of invariant constrained systems using anholonomic frames Next Article A theoretical framework for backward error analysis on manifolds

Clebsch optimal control formulation in mechanics

François Gay-Balmaz^1, and
Tudor S. Ratiu^2,

1.
Centre National de la Recherche Scientifique (CNRS), Laboratoire de Météorologie Dynamique, École Normale Supérieure, Paris
2.
Section de Mathématiques and Bernoulli Center, Ecole Polytechnique Fédérale de Lausanne, Lausanne, CH-1015

Received: September 2010

Revised: April 2011

Published: March 2011

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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.

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Mathematics Subject Classification: 49J15, 49J20, 37K05, 58E30, 70H30.

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