  The controllability of complex networks has received much attention recently,
which tells whether we can steer a system from an initial state to any final
state within finite time with admissible external inputs. In order to
accomplish the control in practice at the minimum cost, we must study how much
control energy is needed to reach the desired final state. At a given control
distance between the initial and final states, existing results present the
scaling behavior of lower bounds of the minimum energy in terms of the control
time analytically. 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 the 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 feasible to the scenario with any
number of nodes that receive inputs directly and any types of networks.
Moreover, more precise analytical results for the lower bound of the minimum
energy are derived in the proposed framework. Our results pave the way to
implement realistic control over various complex networks with the minimum
control cost.
