Controllability of periodic linear systems, the Poincare sphere, and quasi-affine systems

TL;DR

This paper investigates the controllability of periodic linear systems by introducing an autonomous system with phase, utilizing the Poincare sphere for state space compactification, and applying findings to quasi-affine systems.

Contribution

It introduces a novel autonomous system framework for periodic linear control systems and applies the Poincare sphere to analyze behavior at infinity, extending to quasi-affine systems.

Findings

Existence of a unique control set with nonvoid interior for periodic systems.

Use of the Poincare sphere to describe system behavior near infinity.

Application of the method to quasi-affine systems confirming similar control set properties.

Abstract

For periodic linear control systems with bounded control range, an autonomized system is introduced by adding the phase to the state of the system. Here a unique control set (i.e., a maximal set of approximate controllability) with nonvoid interior exists. It is determined by the spectral subspaces of the homogeneous part which is a periodic linear differential equation. Using the Poincar\'e sphere one obtains a compactification of the state space allowing us to describe the behavior near infinity of the original control system. Furthermore, an application to quasi-affine systems yields a unique control set with nonvoid interior.