Symmetries in left-invariant optimal control problems

Left-invariant optimal control problems on Lie groups are considered. When studying the optimality of extreme trajectories, the crucial role is played by symmetries of the exponential map that are induced by symmetries of the conjugate subsystem of the Hamiltonian system of the Pontryagin maximum principle. A general construction is obtained for these symmetries of the exponential map for connected Lie groups with generic coadjoint orbits of codimension not exceeding one and with a connected stabilizer. Bibliography: 32 titles.