Identifying a heat source in automatic control problems

We consider the evolution of the temperature u in a material with thermal memory associated to the problem of keeping the range of u within a given interval of values, on any prescribed finite time interval. To this aim, we formulate an automatic control problem based on a feedback device located on the boundary. Two different types of such a device are analyzed: the relay switch and the Preisach operator. We prove global existence and uniqueness of a solution in both the cases considered. The, assuming that the heat source is partially unknown, we formulate an inverse problem that can be reduced to a system analogous to the previous control problem. Existence and uniqueness results of a solution to the inverse problem are proved.