Control and Integrability on SO (3)

Abstract

This paper considers control ane left- invariant systems evolving on matrix Lie groups. Such systems have signicant applications in a variety of elds. Any left-invariant optimal control problem (with quadratic cost) can be lifted, via the celebrated Maximum Principle, to a Hamiltonian system on the dual of the Lie algebra of the underlying state space G. The (minus) Lie-Poisson structure on the dual space g is used to describe the (normal) extremal curves. An interesting, and rather typical, single-input con- trol system on the rotation group SO (3) is investi- gated in some detail. The reduced Hamilton equa- tions associated with an extremal curve are derived in a simple and elegant manner. Finally, these equations are explicitly integrated by Jacobi elliptic functions.