论文标题
在非平滑椭圆形最佳控制的无空隙二阶最优条件下
On the no-gap second-order optimality conditions for a non-smooth semilinear elliptic optimal control
论文作者
论文摘要
这项工作与二阶必需和足够的最佳条件有关，以最佳控制非平滑椭圆形偏微分方程方程，其中非线性是非平滑的最大功能，因此相关的控制对状态操作员在一般情况下不是GâteAux-ganteaux-differentens。除了站立假设外，还提出了两个主要的假设。第一个是在对目标功能的考虑控制下的gâteaux-差异性，它的特征是在活动集合上消失了伴随状态。第二个是对“几乎”活动集的结构假设，即，感兴趣的状态值“接近”最大功能的非差异性点的所有点的集合。然后，我们根据抽象曲率功能来得出二阶最佳条件的“无差距”理论，即，在这种功能中，必要或足够的二阶优化条件之间唯一的变化是在严格和非严格的不平等之间。
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 active set. The second one is a structural assumption on the 'almost' active sets, i.e., 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 an abstract curvature functional, i.e., for which the only change between necessary or sufficient second-order optimality conditions is between a strict and non strict inequality.
