A.V.Podobryaev. Construction of Maxwell points in left-invariant optimal control problems // Proceedings of Steklov Institute of Mathematics. 315, 190-197 (2021)
We consider left-invariant optimal control problems on connected Lie groups. The Pontryagin maximum principle gives the necessary conditions of optimality. Extremal trajectories are projections of trajectories of the corresponding Hamiltonian system on the cotangent bundle of the Lie group. Maxwell points (i.e., points where two different extremal trajectories meet one another) 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.