The main tool we use in this work is the control theory with partial information. An important quantity in this con- text is the information state, which is a probability vector that weighs the most that can be inferred about the state of the system at a certain time instance, given the system behavior at previous time instances. There are some important results in the literature dealing with related results on convergence in distribution of the information state, in which the state of a system can only be inferred from partial observations. Kaijser proved convergence in distribution of the information state for finite-state ergodic Markov chains, for the case when the chain transition matrix and the function which links the par- tial observation with the original Markov chain (the obser- vation function) satisfy some mild conditions [ 21 ]. Kaijser’s results were used by Goldsmith and Varaiya, in the context of finite-state Markov channels [ 22 ]. This convergence result is obtained as a step in computing the Shannon capacity of finite-state Markov channels, and it holds under the crucial assumption of i.i.d. inputs: a key step of that proof is shown to break down for an example of Markov inputs. This as- sumption is removed in a recent work of Sharma and Singh [ 23 ], where it is shown that for convergence in distribution, the inputs need not be i.i.d., but in turn the pair (channel input, channel state) should be drawn from an irreducible, aperiodic, and ergodic Markov chain. Their convergence re- sult is proved using the more general theory of regenerative processes. However, using directly these results in our setting does not yield the sought result of weak convergence and thus stability, as we will show that the optimal control policy is a function of the information state, whereas in previous work, inputs are independent of the state of the system. This depen- dence due to feedback control is the main di ﬀ erence between our setup and previous work.