李群上的左不变最优控制问题
Left-invariant optimal control problems on Lie groups
摘要：
李群上的左不变最优控制问题是一类重要的大对称群问题。从理论的角度来看，它们是有趣的，因为它们通常可以被完全研究，并且可以在这些模型问题上研究一般特征。特别地，幂零李群上的问题给出了一般问题的基本幂零逼近。左不变问题也经常出现在应用中：在经典和量子力学，几何学，机器人学，视觉模型和图像处理。本文综述了李群上左不变最优控制问题的主要概念、方法和结果。重点讨论了极值轨迹的描述及其最优性、切割时间和割迹、最优综合等问题。书目：238种。
英文摘要：
Left-invariant optimal control problems on Lie groups form an important class of problems with big symmetry group. They are interesting from the theoretical point of view since they often can be completely studied, and general features can be investigated on these model problems. In particular, problems on nilpotent Lie groups give a fundamental nilpotent approximation of general problems. Left-invariant problems also often arise in applications: in classical and quantum mechanics, geometry, robotics, models of vision and image processing. This work aims to review the main notions, methods and results for left-invariant optimal control problems on Lie groups. The main attention is paid to description of extremal trajectories and their optimality, cut time and cut locus, optimal synthesis. Bibliography: 238 titles.
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