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In this note, we study the optimal control of a nonisothermal phase fieldsystem of Cahn-Hilliard type that constitutes an extension of the classicalCaginalp model for nonisothermal phase transitions with a conserved orderparameter. It couples a Cahn-Hilliard type equation with source term for theorder parameter with the universal balance law of internal energy. In place ofthe standard Fourier form, the constitutive law of the heat flux is assumed inthe form given by the theory developed by Green and Naghdi, which accounts fora possible thermal memory of the evolution. This has the consequence that thebalance law of internal energy becomes a second-order in time equation for thethermal displacement or freezing index, that is, a primitive with respect totime of the temperature. Another particular feature of our system is thepresence of the source term in the equation for the order parameter, whichentails further mathematical difficulties because the mass conservation of theorder parameter is no longer satisfied. We study the case that the double-wellpotential driving the evolution of the phase transition is given by thenondifferentiable double obstacle potential, thereby complementing recentresults obtained for the differentiable cases of regular and logarithmicpotentials. We derive first-order necessary optimality conditions for thecontrol problem. The analysis is carried out by employing the so-called deepquench approximation in which the nondifferentiable double obstacle potentialis approximated by a family of logarithmic potentials for which meaningfulfirst-order necessary optimality conditions are available. Since the resultsfor the logarithmic potentials crucially depend on the validity of theso-called strict separation property which is only available in the spatiallytwo-dimensional situation, our whole analysis is restricted to thetwo-dimensional case.

Optimal temperature distribution for a nonisothermal Cahn-Hilliard system in two dimensions with source term and double obstacle potential