Global controllability properties of linear control systems

TL;DR

This work provides a global view of controllability for linear systems with bounded controls by embedding the dynamics into a Poincaré compactification and analyzing the induced control flows. By lifting to a bilinear system in $\\R^{n+1}$ and applying the Selgrade decomposition, the authors relate asymptotic behavior to invariant subbundles and Lyapunov spectra, and then project these structures to the Poincaré sphere to identify chain control sets and limit sets. They establish explicit descriptions of the central and peripheral Selgrade bundles on the sphere, compute Lyapunov exponents in terms of the original eigenvalues of $A$, and construct stable manifolds both on the sphere and in $\\R^{n}$ using a local stable-manifold framework for control flows. The results yield precise information about long-term trajectories, invariant manifolds, and limit behavior at infinity, accompanied by several illustrative examples. These findings offer a rigorous, geometrical handle on global controllability aspects and infinity dynamics for linear control systems under bounded inputs, with potential extensions to affine and polynomial control systems.

Abstract

For linear control systems with bounded control range, the state space is compactified using the Poincaré sphere. The linearization of the induced control flow allows the construction of invariant manifolds on the sphere and of corresponding manifolds in the state space of the linear control system.

Figures (3)

• Figure 1: Phase portraits for $u(t)\equiv0$ in $\mathbb{R}^{3}$ and $\mathbb{S}^{2}\simeq\mathbb{S}^{3,0}$ .
• Figure 2: Phase portraits for $u(t)\equiv0$ in $\mathbb{R}^{3}$ and $\mathbb{S}^{2}\simeq\mathbb{S}^{3,0}$ .
• Figure 3: Phase portraits for $u(t)\equiv0$ in $\mathbb{R}^{3}$ and $\mathbb{S}^{2}\simeq\mathbb{S}^{3,0}$.

Theorems & Definitions (42)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• Theorem 2
• Theorem 3
• proof
• Proposition 1
• Lemma 1
• proof