An asymptotic optimal design for a decentralized system with noisy communication

A reformulation of the Witsenhausen counter-example is considered, where the first station is allowed to transmit its information to the second station through a low noise channel. This is in fact a decentralized stochastic system where the communication uncertainty induces a non-classical information pattern. Assuming a small transmission noise intensity, an asymptotic approach is used in order to find an approximated cost. Then, the necessary conditions for asymptotically optimal strategies are obtained using a variational approach. It is shown that the necessary conditions are satisfied by linear strategies with slightly different coefficients than the noiseless transmission case.