Lie algebroids and optimal control: abnormality

Abstract

Candidates to be solutions to optimal control problems, called extremals, are found using Pontryagin’s Maximum Principle [9]. This Principle gives necessary conditions for optimality and, under suitable assumptions, starts a presymplectic constraint algorithm in the sense given in [3]. This procedure, first considered in optimal control theory in [6], can be adapted to characterize the different kinds of extremals [1].In this paper, we describe the constraints given by the algorithm for the so‐called abnormal extremals for optimal control problems defined on Lie algebroids [4, 7, 8]. The peculiarity of the abnormal extremals is their independence on the cost function to characterize them. In particular, we are interested in how useful the geometry provided by the Lie algebroid is to study the constraints obtained in the optimal control problems for affine connection control systems. These systems model the motion of different types of mechanical systems such as rigid bodies, nonholonomic systems and robotic arms [2].