The flow of messages in a message-switched data communication network is modeled in a continuous dynamical state space. The state variables represent message storage at the nodes and the control variables represent message flow rates along the links. A deterministic linear cost functional is defined which is the weighted total message delay in the network when we stipulate that all the message backlogs are emptied at the final time and the inputs are known. Minimization of the cost functional results in a linear optimal control problem with linear state and control variable inequality constraints. The remainder of the report is devoted to finding the feedback solution to the optimal control problem when all the inputs are constant in time. First, the necessary conditions of optimality are derived and shown to be sufficient. The pointwise minimization in time is a linear program and the optimal control is shown to be of the bang-bang variety. There are several properties of the method which complicate its formulation as a compact algorithm for general network problems. However, in the case of problems involving networks with single destinations and all unity weightings in the cost functional it is shown that these complications do not arise.