具有热记忆的Caginalp型相场模型的分析与最优控制理论
Analysis and optimal control theory for a phase field model of Caginalp type with thermal memory
摘要：
Caginalp相场系统的非线性扩展被认为是考虑到热记忆。由此产生的模型，这是一个一阶近似的一致性系统，是由格林和Naghdi开发的理论的启发。由相动力学和内能的普适平衡定律得出的两个方程，用相变量（代表非守恒序参量）和所谓的热位移来表示，即，一个关于温度时间的基元。证明了相应初边值问题弱解和强解的存在性和连续依赖性。然后，一个最优控制问题进行了研究，一个合适的成本泛函，其中两个数据作为控制，即，分布式热源和初始温度。证明了控制-状态算子在适当的巴拿赫空间之间的Fréchet可微性，并利用包含伴随变量的变分不等式导出了有意义的一阶最优性必要条件.最后，给出了最优控制的特征。
英文摘要：
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. Fréchet 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.
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