Dynamics of linear control systems and stabilization

TL;DR

This paper analyzes linear control systems on connected Lie groups through the lens of drift-induced dynamics. By performing a Jordan decomposition of the drift into elliptic, hyperbolic, and nilpotent parts, it defines dynamical subgroups $G^+$, $G^0$, and $G^-$ and characterizes how bounded positive orbits enforce a global structure $G = G^- G^{+,0}$ with $G^0$ compact, yielding compact control sets and, in the controllable case, a compact state space. It then extends classical stability notions to this non-Euclidean setting, providing precise criteria for internal stability and BIBO stability in terms of the drift’s dynamical subgroups and, for BIBO stability, a homomorphism $F$ with $ ext{ker}F$ invariant and $F(G^0)$ compact. The results unify dynamical systems methods with control-theoretic concepts on Lie groups, offering concrete, verifiable conditions for reachability, stability, and robustness in geometric control applications.

Abstract

In this paper, we study linear control systems with positive bounded orbits. We show that the existence of positive bounded orbits imposes strong algebraic and topological constraints on the state space. In fact, a linear control system has bounded positive orbits if and only if it can be decomposed as the product of the stable and central subgroups of the drift, with the central subgroup being compact. In particular, systems with bounded positive orbits admit a compact control set, and if the system is controllable, the entire state space is a compact group. As a byproduct, we obtain a complete characterization of the internal and BIBO stability of linear control systems.