Analysis and optimal control theory for a phase field model of Caginalp type with thermal memory

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