  We consider a remote estimation problem with an energy harvesting sensor and
a remote estimator. The sensor observes the state of a discrete-time source
which may be a finite state Markov chain or a multi-dimensional linear Gaussian
system. It harvests energy from its environment (say, for example, through a
solar cell) and uses this energy for the purpose of communicating with the
estimator. Due to the randomness of energy available for communication, the
sensor may not be able to communicate all the time. The sensor may also want to
save its energy for future communications. The estimator relies on messages
communicated by the sensor to produce real-time estimates of the source state.
We consider the problem of finding a communication scheduling strategy for the
sensor and an estimation strategy for the estimator that jointly minimize an
expected sum of communication and distortion costs over a finite time horizon.
Our goal of joint optimization leads to a decentralized decision-making
problem. By viewing the problem from the estimator's perspective, we obtain a
dynamic programming characterization for the decentralized decision-making
problem that involves optimization over functions. Under some symmetry
assumptions on the source statistics and the distortion metric, we show that an
optimal communication strategy is described by easily computable thresholds and
that the optimal estimate is a simple function of the most recently received
sensor observation.
