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

Linear Quadratic Gaussian Synthesis for a Heated/Cooled Rod Using Point Actuation and Point Sensing