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

Minimal Actuator Placement with Optimal Control Constraints