  A geometric setup for control theory is presented. The argument is developed
through the study of the extremals of action functionals defined on piecewise
differentiable curves, in the presence of differentiable non-holonomic
constraints. Special emphasis is put on the tensorial aspects of the theory. To
start with, the kinematical foundations, culminating in the so called
variational equation, are put on geometrical grounds, via the introduction of
the concept of infinitesimal control . On the same basis, the usual
classification of the extremals of a variational problem into normal and
abnormal ones is also rationalized, showing the existence of a purely
kinematical algorithm assigning to each admissible curve a corresponding
abnormality index, defined in terms of a suitable linear map. The whole
machinery is then applied to constrained variational calculus. The argument
provides an interesting revisitation of Pontryagin maximum principle and of the
Erdmann-Weierstrass corner conditions, as well as a proof of the classical
Lagrange multipliers method and a local interpretation of Pontryagin's
equations as dynamical equations for a free (singular) Hamiltonian system. As a
final, highly non-trivial topic, a sufficient condition for the existence of
finite deformations with fixed endpoints is explicitly stated and proved.
