  We extend the Pontryagin Maximum Principle (PMP) to the geometric setting of
almost-Lie (AL) algebroids -- objects which generalize Lie algebroids. The
result may be understood as a very general reduction scheme for optimal control
problems (OCPs). It covers the standard PMP, as well as gives necessary
optimality conditions for symmetric OCPs on Lie groups, principal bundles, and
Lie groupoids. We do not assume the symmetry of boundary conditions. The ideas
are based on a very general concept of homotopy of admissible paths on AL
algebroids. Our framework works for OCPs with fixed-end-points and general
boundary conditions.
