Stabilizability Conditions for Linear Time Invariant Systems Across a Gaussian MAC Channel

We consider the problem of stabilizing two scalar plants across a Gaussian multiple access channel (MAC). The states of both the plants are encoded separately and transmitted across a shared MAC channel to a joint decoder/controller. The decoder receives the sum of the two transmissions corrupted by an additive white Gaussian noise (AWGN). A necessary condition for stabilization of the two plants with any encoder and decoder design is obtained. Particular encoder and decoder designs are presented to provide sufficient conditions for stabilization. Two cases are considered: When the encoders can cooperate with each other by sharing information and when they cannot. The results suggest interesting similarities and differences as compared to stabilization of a single plant across a point to point AWGN channel.