Dynamics of linear control systems and stabilization

TL;DR

This paper characterizes linear control systems with positive bounded orbits, revealing their algebraic structure and conditions for stability, and showing that such systems can be decomposed into stable and central components.

Contribution

It provides a complete algebraic and topological characterization of linear control systems with bounded positive orbits, including stability criteria and system decomposition.

Findings

Systems with bounded positive orbits decompose into stable and central subgroups.

Controllable systems with bounded positive orbits have a compact state space.

The paper characterizes internal and BIBO stability for these systems.

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.