Problems of efficient communication by means of the motions of a controlled dynamical system are considered. Application settings in which such problems are of interest include quantum computing and problems of two or more robotic agents communicating with each other by means of their motions, but in the present paper, we restrict our attention to communication between two agents by means of the dynamics of a linear time-invariant control system. The paper considers dual objective control problems in which a primary objective is combined with a communication objective. Solutions of varying degrees of explicitness can be given. Connections with the emerging theory of control communication complexity are discussed.