  We study feedback control of a dynamical process over a lossy channel
equipped with a hybrid automatic repeat request protocol that connects a sensor
to an actuator. The dynamical process is modeled by a Gauss-Markov process, and
the lossy channel by a packet-erasure channel with ideal feedback. We suppose
that data is communicated in the format of packets with negligible quantization
error. In such a networked control system, whenever a packet loss occurs, there
exists a tradeoff between transmitting new sensory information with a lower
success probability and retransmitting previously failed sensory information
with a higher success probability. In essence, an inherent tradeoff between
freshness and reliability. To address this tradeoff, we consider a
linear-quadratic-regulator performance index, which penalizes state deviations
and control efforts over a finite horizon, and jointly design optimal policies
for an encoder and a decoder, which are collocated with the sensor and the
actuator, respectively. Our emphasis here lies specifically on designing
switching and control policies, rather than error-correcting codes. We derive
the structural properties of the optimal encoding and decoding policies. We
show that the former is a threshold switching policy and the latter is a
certainty-equivalent control policy. In addition, we specify the iterative
equations that the encoder and the decoder need to solve in order to implement
the optimal policies.
