The averaged control system of fast oscillating control systems
Résumé
For control systems that either have an explicit periodic dependence on time 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 systems as oscillations go faster. The dimension of its velocity set is characterised geomtrically. When it is maximum the average system defines a Finsler metric, unfortunately not twice differntiable in general. Under particular assumptions, valid for the control two body system, this Finsler metric generates a Hamiltonian flow on the cotangent bundle. For minimum time control, this average system proves that averaging the Hamiltonian given by the maximum principle is a valid approximation.
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