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.

中文翻译：

---

不变式最优控制问题的对称性

考虑Lie群的左不变最优控制问题。当研究极限轨迹的最优性时，由庞特里亚金极大原理的哈密顿系统的共轭子系统的对称性引起的指数图的对称性起着至关重要的作用。对于连接的李群的指数图的这些对称性，获得了一个一般的构造，这些李群具有共维的共共伴轨道不超过一个，并且具有一个稳定子。参考书目：32种。