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This paper explores the problem of feedback coding for a channel whose output is simultaneously used for two purposes: it is decoded to establish reliable communication, and it is used to control a dynamical system. In general, there is a tradeoff between the rate of communication and the accuracy of control. An intuitive communication and control strategy is analyzed and shown to be optimal in several cases of interest. Using methods from stochastic control, a corresponding upper bound is derived on the rate for a given cost, and this bound can be applied to find the capacity for an interesting class of channels. Under certain regularity conditions, the capacity has a simple characterization, and a series of examples is provided to demonstrate how it can be calculated.