  The fundamental theorem of the theory of optimal control, the Pontryagin
maximum principle (PMP), is extended to the setting of almost Lie (AL)
algebroids, geometrical objects generalizing Lie algebroids. This formulation
of the PMP yields, in particular, a scheme comprising reductions of optimal
control problems similar to the reduction for the rigid body in analytical
mechanics. On the other hand, in the presented approach the reduced and
unreduced PMPs are parts of the same universal formalism. The framework is
based on a very general concept of homotopy of measurable paths and the
geometry of AL algebroids.
