In some cases the most important factor limiting the performance of a distributed control system is not the availability of computational power but rather the availablity of time on a shared communication network for communication between the sensors, the control computer and the actuators. In this paper the author develops a mathematical model describing a class of multivariable control problems of this type and presents an algorithm which can be used to investigate the existence of stabilizing control laws in the presence of communication constraints. The author's model assumes that the communication facilities are to be time-shared according to a pattern which is repeated periodically. The designer has the problem of picking the pattern such that effective control laws can be implemented within the constraints it imposes. If the systems being controlled are linear, there is an affine family of possible closed-loop transition matrices associated with each communication pattern. The selection and implementation of a particular control law which is supported by the given pattern then defines the performance. This approach allows us to put the problem of designing the communication pattern in a form that can be investigated using mathematical programming techniques. In particular one can evaluate the advantages and disadvantages associated with allocating more communication resources to some control loops and less to others.