The averaged control system of fast-oscillating control systems (Q2848585)

For control systems that either have a fast explicit periodic dependence on time and bounded controls or have periodic solutions and small controls, an average control system that takes into account all possible variations of the control is designed. It is proved 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 averaged system defines a Finsler metric, which is not twice differentiable in general. For minimum time control, the averaged system allows one to give a rigorous proof that averaging the Hamiltonian given by the maximum principle is a valid approximation. The general methodology is applied to Kepler control systems and also to the minimum time orbit transfer in the planar 2-body problem.