This work is concerned with second-order necessary and sufficient optimality conditions for optimal control of a non-smooth semilinear elliptic partial differential equation, where the nonlinearity is the non-smooth max-function and thus the associated control-to-state operator is, in general, not Gâteaux-differentiable. In addition to standing assumptions, two main hypotheses are imposed. The first one is the Gâteaux-differentiability at the considered control of the objective functional and it is precisely characterized by the vanishing of an adjoint state on the set of all zeros of the corresponding state. The second one is a structural assumption on the sets of all points at which the values of the interested state are ‘close’ to the non-differentiability point of the max-function. We then derive a ‘no-gap’ theory of second-order optimality conditions in terms of a second-order generalized derivative of the cost functional, i.e. for which the only change between necessary and sufficient second-order optimality conditions are between a strict and non-strict inequality.

这项工作涉及非光滑半线性椭圆偏微分方程最优控制的二阶必要和充分最优性条件，其中非线性是非光滑最大函数，因此相关的控制状态算子是，一般来说，不是 Gâteaux 可微的。除了常设假设外，还强加了两个主要假设。第一个是目标泛函的考虑控制下的 Gâteaux 可微性，它的精确特征在于相应状态的全零集合上的伴随状态消失。第二个是对感兴趣状态的值“接近”最大函数的不可微分点的所有点的集合的结构假设。