  We introduce the problem of minimal actuator placement in a linear control
system so that a bound on the minimum control effort for a given state transfer
is satisfied while controllability is ensured. We first show that this is an
NP-hard problem following the recent work of Olshevsky. Next, we prove that
this problem has a supermodular structure. Afterwards, we provide an efficient
algorithm that approximates up to a multiplicative factor of O(logn), where n
is the size of the multi-agent network, any optimal actuator set that meets the
specified energy criterion. Moreover, we show that this is the best
approximation factor one can achieve in polynomial-time for the worst case.
Finally, we test this algorithm over large Erdos-Renyi random networks to
further demonstrate its efficiency.
