Feedback stabilization of a marginally stable heat conduction problem

Abstract

Semigroup methods are used to study a class of feedback controls for exponential stabilization of the (marginally stable) standard heat equation in a region with insulated boundary. An example shows that seemingly reasonable feedbacks may lead to instabilities if the gain factor is too large. We compute an upper bound for the gain factors in the stable range, and also state and support a conjecture corresponding to a case when the feedback elements have a restricted sign. Generalizations in various directions are discussed.