  In the present paper, we present and solve the sliding mode control (SMC)
problem for a second-order generalization of the Caginalp phase-field system.
This generalization, inspired by the theories developed by Green and Naghdi on
one side, and Podio-Guidugli on the other, deals with the concept of thermal
displacement, i.e., a primitive with respect to the time of the temperature.
Two control laws are considered: the former forces the solution to reach a
sliding manifold described by a linear constraint between the temperature and
the phase variable; the latter forces the phase variable to reach a prescribed
distribution $\varphi^*$. We prove existence, uniqueness as well as continuous
dependence of the solutions for both problems; two regularity results are also
given. We also prove that, under suitable conditions, the solutions reach the
sliding manifold within finite time.

Analysis of PDEs