Reachable Volume of Large-Scale Linear Network Systems: The Single-Input Case

The volume of the set of states of a linear system that can be reached using control inputs with bounded energy depends on the determinant of the controllability Gramian. As such, the latter represents a meaningful metric to quantify the degree of controllability of a linear system. In this paper, we conduct a detailed study of this controllability metric for the case of linear network systems controlled by a single node. Our analysis builds on a closed-form expression which uncovers the relation between the determinant of the controllability Gramian and the spectral properties of the underlying system. In particular, we explore the dependence of this metric on the network dimension and provide a characterization of its scaling behavior in terms of exact exponential decay rates. We illustrate and complement our findings with a set of numerical examples.