Optimizing controllability metrics for target controllability

While dealing with the problem of control of complex networks, in addition to verifying qualitative properties of whether the system is controllable or not, one needs to quantify the effort needed to control the system. This is because the required control effort becomes significantly large, especially when there are constraints on the number of control inputs, rendering the system practically uncontrollable. In some cases, it may not be required to control all the nodes of the network but rather a subset of states called target nodes, in which case the energy requirements reduce substantially with dropping off few nodes for control. Building upon this finding, we attempt to solve three problems in this paper. First, using the average controllability as a metric, we identify the best set of $p$ target nodes that maximize the average controllability. In practical situations, one needs to know an upper bound on the input energy. Our second problem identifies the largest set of target nodes given worst case energy bound, using the minimum eigenvalue of the gramian as the metric. Lastly, given the size of a target set, we aim to identify the set of nodes that minimize the upper bound of worst case energy needed for control. We validate our findings on some tractable examples and randomly generated Erdos-Renyi Networks.