The paper is concerned with a new class of differential hemivariational inequalities which appears as the weak formulation of steady thermistor problems with mixed boundary conditions. First, we show the existence of solution to this kind of inequality problems combining the theory of pseudomonotone operators and a fixed point argument. Then, an optimal control problem is considered where the control is represented by the heat source. We introduce parameter perturbations of the electric conductivity and the boundary temperature in the system to examine their impact on the sensitivity properties of the optimal control problem. We prove that the optimal state-control set is nonempty and the value function of the optimal control problem is continuous. Finally, the multivalued map induced by optimal state-control set is established to be weakly upper semicontinuous in the weak topology.

中文翻译：

---

稳态热敏电阻问题中差分半变分不等式最优控制问题的敏感性分析


本文关注一类新的差分半变分不等式，它表现为混合边界条件下稳态热敏电阻问题的弱公式。首先，我们结合伪单调算子理论和不动点论证，展示了这种不等式问题的解决方案的存在。然后，在控制由热源表示的情况下，考虑最优控制问题。我们引入了系统中电导率和边界温度的参数扰动，以检查它们对最优控制问题的灵敏度特性的影响。我们证明了最优状态控制集是非空的，并且最优控制问题的值函数是连续的。最后，将最优状态控制集诱导的多值映射建立为弱上半连续在弱拓扑中。