Energy cost for controlling complex networks with linear dynamics

Examining the controllability of complex networks has received much attention recently. The focus of many studies is commonly trained on whether we can steer a system from an arbitrary initial state to any final state within finite time with admissible external inputs. In order to accomplish the control at the minimum cost, we must study how much control energy is needed to reach the desired state. At a given control distance between the initial and final states, existing results have offered the scaling behavior of lower bounds of the minimum energy in terms of the control time. However, to reach an arbitrary final state at a given control distance, the minimum energy is actually dominated by the upper bound, whose analytic expression still remains elusive. Here we theoretically show the scaling behavior of a precise upper bound of the minimum energy in terms of the time required to achieve control. Apart from validating the analytical results with numerical simulations, our findings are applicable to any number of nodes that receive inputs directly and any types of networks with linear dynamics. Moreover, more precise analytical results for the lower bound of the minimum energy are derived with the proposed method. Our results pave the way for implementing realistic control over various complex networks with the minimum control cost.