We study a class of optimal control problems that are encountered in consensus and synchronization networks. These are characterized by structural constraints that arise from the lack of the absolute measurements for a part of the state vector. In order to deal with these constraints, we introduce a coordinate transformation to eliminate the average mode and assure stabilizability and detectability of the remaining modes. To design networks with low communication requirements, we seek solutions to the ℓ1 regularized version of the standard 2 optimal control problem. Such solutions trade off network performance and sparsity of the controller. We also identify a class of optimal control problems that can be cast as semidefinite programs. Examples from power systems and consensus networks are provided to illustrate our developments.