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 the frequency of the oscillations tends to infinity. The dimension of its velocity set is characterized geometrically. When it is maximum the average system defines a Finsler metric, which is 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.