The averaged control system of fast oscillating control systems
Provides a theoretical foundation for averaging in control systems with fast oscillations, benefiting researchers in optimal control and dynamical systems.
The paper defines an averaged control system for fast oscillating systems and proves its solutions approximate those of the original system as oscillations increase. It characterizes the dimension of the velocity set and shows the averaged system yields a Finsler metric, enabling rigorous proof that Hamiltonian averaging from the maximum principle is valid for minimum time control.
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.