Construction of Maxwell Points in Left-Invariant Optimal Control Problems

We consider left-invariant optimal control problems on connected Lie groups. The Pontryagin maximum principle gives necessary optimality conditions. Namely, the extremal trajectories are the projections of trajectories of the corresponding Hamiltonian system on the cotangent bundle of the Lie group. The Maxwell points (i.e., the points where two different extremal trajectories meet each other) play a key role in the study of optimality of extremal trajectories. The reason is that an extremal trajectory cannot be optimal after a Maxwell point. We introduce a general construction for Maxwell points depending on the algebraic structure of the Lie group.