Cubic polynomials and optimal control on compact Lie groups

Abstract

This paper analyzes the Riemannian cubic polynomials’s problem from a Hamiltonian point of view. The description of the problem on compact Lie groups is particulary explored. The state space of the second order optimal control problem considered is the tangent bundle of the Lie group which also has a group structure. The dynamics of the problem is described by a presymplectic formalism associated with the canonical symplectic form on the cotangent bundle of the tangent bundle. Using these control geometrical tools, the equivalence between the Hamiltonian approach developed here and the known variational one is verified. Moreover, the equivalence allows us to deduce two invariants along the cubic polynomials which are in involution.