  For control systems that either have a fast explicit periodic dependence on
time and bounded controls or have periodic solutions and small controls, we
define an average control system that takes into account all possible
variations of the control, and prove that its solutions approximate all
solutions of the oscillating system as oscillations go faster. The dimension of
its velocity set is characterized geometrically. When it is maximum the average
system defines a Finsler metric, not twice differentiable in general. For
minimum time control, this average system allows one to give a rigorous proof
that averaging the Hamiltonian given by the maximum principle is a valid
approximation.
