Optimality on homogeneous spaces, and the angle system associated with a bilinear control system

Let G be a Lie group. In order to study optimal control problems on a homogeneous space G/H, we identify its cotangent bundle T^∗G/H as a subbundle of the cotangent bundle of G. Next, this identification is used to describe the Hamiltonian lifting of vector fields on G/H induced by elements in the Lie algebra g of G. As an application, we consider a bilinear control system Σ in ℝ² whose matrices generate sl(2). Through the Pontryagin maximum principle, we analyze the time-optimal problem for the angle system PΣ defined by the projection of Σ onto the projective line P¹. We compute some examples, and in particular we show that the bang-bang principle does not need to be true.