A formula for backward and control problems of the heat equation

Qi S Zhang

Published 2020-05-17Version 1

Using time analyticity result, a new perspective for some basic questions for a nonhomogeneous backward heat equation (exact control problem) is given in the setting of smooth domains and compact manifolds. Comparing with the classical results \cite{LR:1} and \cite{FI:1}, there are two developments if the full domain is used for control. One is that to reach the same final state as the time dependent controls, the control function (nonhomogeneous term) can be essentially independent of time, i.e. it is 0 on one time interval and stationary on the other. The other is that an explicit formula for the control function is found in the form of an infinite series involving the heat kernel. A byproduct is an inversion formula for the heat kernel.