  This paper considers control systems defined on Lie algebroids. After
deriving basic controllability tests for general control systems, we specialize
our discussion to the class of mechanical control systems on Lie algebroids.
This class of systems includes mechanical systems subject to holonomic and
nonholonomic constraints, mechanical systems with symmetry and mechanical
systems evolving on semidirect products. We introduce the notions of linear
connection, symmetric product and geodesically invariant subbundle on a Lie
algebroid. We present appropriate tests for various notions of accessibility
and controllability, and analyze the relation between the controllability
properties of control systems related by a morphism of Lie algebroids.
