This work is concerned with an optimal control problem governed by a non-smooth quasilinear elliptic equation with a nonlinear coefficient in the principal part that is locally Lipschitz continuous and directionally but not Gâteaux differentiable. This leads to a control-to-state operator that is directionally but not Gâteaux differentiable as well. Based on a suitable regularization scheme, we derive C- and strong stationarity conditions. Under the additional assumption that the nonlinearity is a $ PC^1 $ function with countably many points of nondifferentiability, we show that both conditions are equivalent. Furthermore, under this assumption we derive a relaxed optimality system that is amenable to numerical solution using a semi-smooth Newton method. This is illustrated by numerical examples.

中文翻译：

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一类非光滑拟线性椭圆方程的最优控制

这项工作涉及一个最优控制问题，该问题由一个非光滑的拟线性椭圆方程组控制，该方程组的主要部分具有非线性系数，该局部方程式是局部Lipschitz连续且有方向性的，但Gâteaux不可微。这导致了一个控制到状态的算子，它是有方向的，但也不是 Gâteaux 可微的。基于合适的正则化方案，我们推导出 C 和强平稳条件。在非线性是具有可数许多不可微分点的 $ PC^1 $ 函数的附加假设下，我们证明这两个条件是等价的。此外，在这个假设下，我们导出了一个宽松的最优系统，该系统适合使用半光滑牛顿法进行数值求解。这可以通过数值例子来说明。