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 Fréchet 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. Crucial for the whole analysis to work is the boundedness property stating that the order parameter attains its values in a compact subset of the interior of the effective domain of the nonlinearity. While this property turns out to be generally valid for regular potentials in three dimensions of space, it can be shown for the logarithmic case only in two dimensions.

在本文中，我们研究了 Cahn-Hilliard 型非等温相场系统的最优控制，该系统构成了具有保守序参数的非等温相变经典 Caginalp 模型的扩展。该系统将带有序参数源项的 Cahn-Hilliard 型方程与内能通用平衡定律耦合起来。代替标准傅立叶形式，热通量的本构定律被假定为格林和纳格迪开发的理论给出的形式，这解释了演化的可能的热记忆。这导致内能平衡定律变成热位移或冻结指数的二阶时间方程，即温度相对于时间的原语。我们系统的另一个特殊特征是阶次参数方程中存在源项，这会带来额外的数学困难，因为不再满足经典卡恩-希利亚德方程典型的阶次参数的质量守恒。在本文中，我们分析了驱动相变演化的双阱势在整个实数集上（在常规情况下）或（在奇异对数情况下）在有限开区间上可微的情况; 不可微分的情况（如双障碍电位）被排除在分析之外。我们证明了控制到状态算子在规则和对数情况下的适当 Banach 空间之间的 Fréchet 可微性，并建立了相应伴随系统的可解性，以便导出最优控制的相关一阶必要最优性条件问题。整个分析工作的关键是有界性属性，该属性表明阶次参数在非线性有效域内部的紧凑子集中获得其值。虽然该属性对于三维空间中的常规势通常有效，但它只能在二维空间中的对数情况下显示。