  A nonlinear extension of the Caginalp phase field system is considered that
takes thermal memory into account. The resulting model, which is a first-order
approximation of a thermodynamically consistent system, is inspired by the
theories developed by Green and Naghdi. Two equations, resulting from phase
dynamics and the universal balance law for internal energy, are written in
terms of the phase variable (representing a non-conserved order parameter) and
the so-called thermal displacement, i.e., a primitive with respect to time of
temperature. Existence and continuous dependence results are shown for weak and
strong solutions to the corresponding initial-boundary value problem. Then, an
optimal control problem is investigated for a suitable cost functional, in
which two data act as controls, namely, the distributed heat source and the
initial temperature. Fr\'echet differentiability between suitable Banach spaces
is shown for the control-to-state operator, and meaningful first-order
necessary optimality conditions are derived in terms of variational
inequalities involving the adjoint variables. Eventually, characterizations of
the optimal controls are given.
