  We consider a rod that is heated/cooled and sensed at multiple point
locations. To stabilize it to a constant temperature we set up a Linear
Quadratic Regulator that we explicitly solve by the method of completing the
square to find the optimal linear state feedback for the point actuators. But
we don't assume that the whole state is mesureable so we construct an infinite
dimensional Kalman filter to estimate the whole state from a finite number of
noisy point measurements. These two components yield a Linear Quadratic
Gaussian (LQG) Synthesis for the heat equation under point actuation and point
sensing.
