In the present paper, we present and solve the sliding mode control (SMC) problem for a second-order generalization of the Caginalp phase-field system. This generalization, inspired by the theories developed by Green and Naghdi on one side, and Podio-Guidugli on the other, deals with the concept of thermal displacement, i.e., a primitive with respect to the time of the temperature. Two control laws are considered: the former forces the solution to reach a sliding manifold described by a linear constraint between the temperature and the phase variable; the latter forces the phase variable to reach a prescribed distribution \(\varphi ^*\). We prove existence, uniqueness as well as continuous dependence of the solutions for both problems; two regularity results are also given. We also prove that, under suitable conditions, the solutions reach the sliding manifold within finite time.

在本文中，我们提出并解决了针对Caginalp相场系统的二阶泛化的滑模控制（SMC）问题。这一概括受到一方面是格林和纳格迪（Green and Naghdi）提出，另一方面是波迪奥-古迪格里（Podio-Guidugli）提出的理论的启发，涉及热位移的概念，即关于温度时间的原始形式。考虑了两个控制定律：前一个定律迫使溶液到达一个由温度和相位变量之间的线性约束描述的滑动歧管；后者迫使相位变量达到规定的分布\（\ varphi ^ * \）。我们证明了这两个问题的存在性，唯一性以及对解决方案的持续依赖；还给出了两个规律性结果。我们还证明，在合适的条件下，解可以在有限的时间内到达滑动歧管。