Worst-case control energy of single-input network systems

The controllability of large-scale network systems has been extensively investigated in the past few years. In spite of the recent advances in this field, there are still a number of unsolved problems which are of critical importance to fully understand the practical limitations arising in the control of large-scale networks. These include the derivation of informative bounds and scaling laws for the control energy of networks controlled by a limited number of nodes. In this paper, we aim to fill this gap by establishing new numerically reliable bounds and asymptotic estimates on the worst-case control energy of continuous-Time linear network systems controlled by a single node. Our results rely on a convenient reformulation of the controllability Gramian of a single-input linear system in terms of a Cauchy matrix. We illustrate and validate our theoretical findings through several examples, ranging from structured networks to random ones.