Complete lift of control system

TL;DR

This paper investigates the properties of complete lifts of affine control systems on manifolds, revealing that while controllability of the original system implies chain controllability of the lift, the lift itself is never fully controllable.

Contribution

It provides an explicit geometric description of the lifted system and introduces chain controllability as a key concept linking the controllability of original and lifted systems.

Findings

Complete lifts are never controllable due to geometric constraints.

Controllability of the original system implies chain controllability of the lift.

An explicit geometric description of solutions of the lifted system is provided.

Abstract

We study affine control systems on smooth manifolds and their complete lifts to the tangent bundle, providing an explicit geometric description of the solutions of the lifted system. We show that, although controllability of the complete lift implies controllability of the original system, the lifted system is never controllable due to intrinsic geometric constraints. By introducing chain controllability, we prove that controllability of the original system guarantees chain controllability of its complete lift.