  In this paper consider a two user multiple access channel with noisy
feedback. There are two senders with independent messages who transmit symbols
across an additive white Gaussian channel to a receiver, who in turn sends back
a symbol which is received by the two senders through two independent noisy
Gaussian channels. We consider the case when the feedback is active i.e. the
receiver actively encodes the feedback using a linear state process. We pose
this as a problem of linear sequential coding at the senders and the receiver
to minimize the terminal mean square probability of error at the receiver. This
is an instance of decentralized control with no common information at the
senders and the receiver. In this paper, we construct two linear controllers at
the sender and the receiver. Due to linearity of the policies and the
controllers, all the random variables involved are jointly Gaussian. Moreover,
the corresponding covariance matrix at the receiver of the estimation process
of the senders' messages is a deterministic process, which is a function of the
parameters of the controllers and the strategies of the players, and is thus
perfectly observed by the senders. Based on this observation, we use
deterministic dynamic programming to find the optimal policies and the optimal
linear controllers at both the senders and the receiver. The problem with
passive feedback can be considered as a special case.
