Optimal Control Problems for Affine Connection Control Systems: Characterization of Extremals

摘要

Pontryagin’s Maximum Principle [8] is considered as an outstanding achievement of\noptimal control theory. This Principle does not give sufficient conditions to compute an optimal\ntrajectory; it only provides necessary conditions. Thus only candidates to be optimal trajectories,\ncalled extremals, are found. Maximum Principle gives rise to different kinds of them and, particularly,\nthe so-called abnormal extremals have been studied because they can be optimal, as Liu and\nSussmann, and Montgomery proved in subRiemannian geometry [5, 7].\nWe build up a presymplectic algorithm, similar to those defined in [2, 3, 4, 6], to determine\nwhere the different kinds of extremals of an optimal control problem can be. After describing such\nan algorithm, we apply it to the study of extremals, specially the abnormal ones, in optimal control\nproblems for affine connection control systems [1]. These systems model the motion of different\ntypes of mechanical systems such as rigid bodies, nonholonomic systems and robotic arms [1].