  We consider linear feedback control of the two-dimensional flow past a
cylinder at low Reynolds numbers, with a particular focus on the optimal
placement of a single sensor and a single actuator. To accommodate the high
dimensionality of the flow we compute its leading resolvent forcing and
response modes to enable the design of H2-optimal estimators and controllers.
We then investigate three control problems: i) optimal estimation (OE) in which
we measure the flow at a single location and estimate the entire flow; ii)
full-state information control (FIC) in which we measure the entire flow but
actuate at only one location; and iii) the overall feedback control problem in
which a single sensor is available for measurement and a single actuator is
available for control. We characterize the performance of these control
arrangements over a range of sensor and actuator placements and discuss
implications for effective feedback control when using a single sensor and a
single actuator. The optimal sensor and actuator placements found for the OE
and FIC problems are also compared to those found for the overall feedback
control problem over a range of Reynolds numbers. This comparison reveals the
key factors and conflicting trade-offs that limit feedback control performance.
