  We study a real-time tracking problem in an energy harvesting status update
system with a Markov source and an imperfect channel, considering both sampling
and transmission costs. The problem primary challenge stems from the
non-observability of the source due to the sampling cost. By using the age of
incorrect information (AoII) as a semantic-aware performance metric, our main
goal is to find an optimal policy that minimizes the time average AoII subject
to an energy-causality constraint. To this end, a stochastic optimization
problem is formulated and solved by modeling it as a partially observable
Markov decision process (POMDP). More specifically, to solve the main problem,
we use the notion of a belief state and cast the problem as a belief MDP
problem. Then, for the perfect channel setup, we effectively truncate the
corresponding belief space and solve the MDP problem using the relative value
iteration method. For the general setup, a deep reinforcement learning policy
is proposed. The simulation results show the efficacy of the derived policies
in comparison to an AoI-optimal policy and an opportunistic baseline policy.
