Optimal Control of Cluster Consensus in Networks with Equitable Partitions

In this paper, we tackle the problem of controlling the cluster consensus state corresponding to an equitable partition in linear networks described by the pair (A, B). The matrix A describes the structure of a directed graph and the matrix B defines the nodes in which control input signals are injected. We show that the presence of equitable partitions corresponds to the existence of an A-invariant subspace, the cluster consensus subspace, along which nodes belonging to the same cluster follow the same time evolution. We show that this subspace encompasses the network controllable subspace. Under the assumptions that the dynamics along the cluster consensus subspace is controllable, we design optimal control actions to control the cluster consensus solution either when the initial condition lies on the cluster consensus subspace or when the dynamics transverse to this subspace is asymptotically stable.