  We consider a status update system consisting of a finite-state Markov
source, an energy-harvesting-enabled transmitter, and a sink. The forward and
feedback channels between the transmitter and the sink are error-prone. We
study the problem of minimizing the long-term time average of a (generic)
distortion function subject to an energy causality constraint. Since the
feedback channel is error-prone, the transmitter has only partial knowledge
about the transmission results and, consequently, about the estimate of the
source state at the sink. Therefore, we model the problem as a partially
observable Markov decision process (POMDP), which is then cast as a belief-MDP
problem. The infinite belief space makes solving the belief-MDP difficult.
Thus, by exploiting a specific property of the belief evolution, we truncate
the state space and formulate a finite-state MDP problem, which is then solved
using the relative value iteration algorithm (RVIA). Furthermore, we propose a
low-complexity transmission policy in which the belief-MDP problem is
transformed into a sequence of per-slot optimization problems. Simulation
results show the effectiveness of the proposed policies and their superiority
compared to a baseline policy. Moreover, we numerically show that the proposed
policies have switching-type structures.
