  In this note, we study the optimal control of a nonisothermal phase field
system of Cahn-Hilliard type that constitutes an extension of the classical
Caginalp model for nonisothermal phase transitions with a conserved order
parameter. The system couples a Cahn-Hilliard type equation with source term
for the order parameter with the universal balance law of internal energy. In
place of the standard Fourier form, the constitutive law of the heat flux is
assumed in the form given by the theory developed by Green and Naghdi, which
accounts for a possible thermal memory of the evolution. This has the
consequence that the balance law of internal energy becomes a second-order in
time equation for the thermal displacement or freezing index, that is, a
primitive with respect to time of the temperature. Another particular feature
of our system is the presence of the source term in the equation for the order
parameter, which entails additional mathematical difficulties because the mass
conservation of the order parameter, typical of the classic Cahn-Hilliard
equation, is no longer satisfied. In this paper, we analyze the case that the
double-well potential driving the evolution of the phase transition is
differentiable, either (in the regular case) on the whole set of reals or (in
the singular logarithmic case) on a finite open interval; nondifferentiable
cases like the double obstacle potential are excluded from the analysis. We
prove the Fr\'echet differentiability of the control-to-state operator between
suitable Banach spaces for both the regular and the logarithmic cases and
establish the solvability of the corresponding adjoint systems in order to
derive the associated first-order necessary optimality conditions for the
optimal control problem.
