  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.
