Controllability of induced bilinear systems on the sphere

TL;DR

This paper studies the controllability of bilinear control systems on the sphere, establishing conditions under which the system can be driven from any initial to any final state.

Contribution

It proves that a specific algebraic condition guarantees controllability of bilinear systems on the sphere, providing explicit trajectories for state transfer.

Findings

Lie algebra rank condition is satisfied when [A, B] ≠ 0

The system is controllable under the algebraic condition

Explicit trajectories demonstrate controllability

Abstract

In this paper, we investigate the controllability of bilinear control systems of the form $\dot{s} = As + uBs$, where $s \in \mathbb{S}^2$ and $A, B \in gl(3, \mathbb{R})$ are skew-symmetric matrices. First, we prove that the algebraic condition $[A, B] \neq 0$ ensures that the Lie algebra rank condition is satisfied for these systems. Next, we show that this same condition implies the controllability of the system. Finally, in an explicit and descriptive manner, we demonstrate controllability by exhibiting trajectories that transfer a given initial state to another.