Asymptotic control theory for a system of linear oscillators

TL;DR

This paper develops an asymptotic control strategy for multiple linear oscillators with a common bounded control, proving its near-optimality for large initial energies using advanced mathematical theories.

Contribution

It introduces a new asymptotic control design method for linear oscillators and proves its correctness and near-optimality using DiPerna-Lions theory and perturbation theory.

Findings

Control law is asymptotically optimal for large energies.

Proved the control law correctly defines system motion.

Introduced a new perturbation theory for observable linear systems.

Abstract

We present an asymptotic control theory for a system of an arbitrary number of linear oscillators under a common bounded control. We suggest a design method of a feedback control for this system. By using the DiPerna-Lions theory of singular ODEs, we prove that the suggested control law correctly defines the motion of the system. The obtained control is asymptotically optimal: the ratio of the motion time to zero under this control to the minimum one is close to 1 if the initial energy of the system is large. The results are partially based on a new perturbation theory of observable linear systems.