A discrete time stochastic feedback control system consisting of a nonlinear plant, a sensor, a controller, and a noisy communication channel between the sensor and the controller is considered. The sensor has limited memory and, at each time, it transmits an encoded symbol over the channel and updates its memory. The controller receives a noise-corrupted copy of the transmitted symbol. It generates a control action based on all its past observations and all its past actions. This control action is fed back to the plant. At each time instant the system incurs an instantaneous cost depending on the state of the plant and the control action. The objective is to choose encoding, memory update, and control strategies to minimize an expected total cost over a finite horizon, or an expected discounted cost over an infinite horizon, or an average cost per unit time over an infinite horizon. A solution methodology for obtaining a sequential decomposition of the global optimization problem is developed. This solution methodology is extended to the case when the sensor makes an imperfect observation of the state of the plant.