Zero forcing, linear and quantum controllability for systems evolving on networks

TL;DR

This paper explores the relationship between graph theory, linear control, and quantum controllability, demonstrating how zero forcing sets influence the controllability of networked systems and connecting classical and quantum perspectives.

Contribution

It establishes that zero forcing sets ensure controllability in network systems and links quantum Lie algebraic controllability with classical linear control conditions.

Findings

Zero forcing sets guarantee controllability of network systems.

A connection between quantum and classical controllability is demonstrated.

Zero forcing offers a new tool for analyzing complex networks.

Abstract

We study the dynamics of systems on networks from a linear algebraic perspective. The control theoretic concept of controllability describes the set of states that can be reached for these systems. Under appropriate conditions, there is a connection between the quantum (Lie theoretic) property of controllability and the linear systems (Kalman) controllability condition. We investigate how the graph theoretic concept of a zero forcing set impacts the controllability property. In particular, we prove that if a set of vertices is a zero forcing set, the associated dynamical system is controllable. The results open up the possibility of further exploiting the analogy between networks, linear control systems theory, and quantum systems Lie algebraic theory. This study is motivated by several quantum systems currently under study, including continuous quantum walks modeling transport phenomena. Additionally, it proposes zero forcing as a new notion in the analysis of complex networks.