A separation principle for linear parabolic equations

The classical separation principle is revisited here for a class of linear unstable parabolic equations with distributed input and boundary output. It is proved that, if the open-loop equation owns a single unstable mode and the corresponding positive eigenvalue has a moderate magnitude, the dynamics can be stabilized by means of a dynamic output feedback defined through a suitable observer. Accordingly, the established separation principle allows for selecting the control gains independently from the output injection gain in the observer.