Consensus with linear objective maps

We consider in this paper a weighted consensus problem over directed graphs, where the agents interact in a decentralized way, with the resulting dynamics formulated in continuous time. We introduce a linear function, called the objective map, that defines the desired final state as a convex combination of the initial states of the agents. We provide a complete answer to the question of whether there is a decentralized consensus dynamics over a given digraph which converges to the final state specified by an objective map. In particular, we characterize not only the set of objective maps that are feasible for a given digraph, but also the consensus dynamics that implements the objective map.