The main purpose is the study of optimal control problem in a domain with rough boundary for the mixed Dirichlet‐Neumann boundary value problem for the strongly nonlinear elliptic equation with exponential nonlinearity. A density of surface traction u acting on a part of rough boundary is taken as a control. The optimal control problem is to minimize the discrepancy between a given distribution and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for a given control. After having defined a suitable functional class in which we look for solutions, we prove the consistency of the original optimal control problem and show that it admits a unique optimal solution. Then we derive a first‐order optimality system assuming the optimal solution is slightly more regular.

中文翻译：

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具有粗糙边界的区域中不适定椭圆方程的最优边界控制问题。存在结果和最优条件

主要目的是研究具有指数非线性的强非线性椭圆方程的混合Dirichlet-Neumann边值问题的具有粗糙边界的区域中的最优控制问题。表面牵引密度u作用于粗糙边界的一部分作为控制。最佳控制问题是使给定分布与当前系统状态之间的差异最小化。当我们不能期望给定控制具有状态方程的解时，我们将处理这种非线性情况。在定义了适合的功能类并在其中寻找解决方案之后，我们证明了原始最优控制问题的一致性，并表明它接受了唯一的最优解决方案。然后我们推导一阶最优系统，假设最优解的规则性更高。