The averaged control system of fast oscillating control systems

TL;DR

This paper introduces an average control system for fast oscillating control systems, demonstrating that its solutions approximate the original system's solutions as oscillations increase, with applications to minimum time control.

Contribution

It defines an average control system for oscillating systems, characterizes its velocity set geometrically, and proves the validity of averaging the Hamiltonian for minimum time control.

Findings

Solutions of the average system approximate those of the oscillating system as oscillations increase.

The velocity set's dimension is characterized geometrically.

A rigorous proof is provided for the validity of averaging the Hamiltonian in minimum time control.

Abstract

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.