H∞ Control of a Heated/Cooled Rod Under Point Actuation and Point Sensing

Our long term goal is to extend the H
^∞
paradigm to nonlinear, infinite dimensional systems under point actuation and point sensing. This paper is the first step in this direction. We find the minimum L
²
gain from the noisy heat flux at one end of a rod to the temperature at an arbitrary point of the rod that can be achieved by state feedback on the heat flux at the other end of the rod. We do this by completing the square to obtain a Riccati partial differential equation whose solution yields a state feedback control law that achieves a given L
²
gain. We solve the Riccati PDE by Fourier series. We iterate this process to find the minimum L
²
gain.