论文标题
非平滑准线性椭圆控制问题的数值分析：I。明确的二阶最佳条件
Numerical analysis of a nonsmooth quasilinear elliptic control problem: I. Explicit second-order optimality conditions
论文作者
论文摘要
在本文中，我们将局部最小化器的显式二阶必需和足够的最佳条件引入到准线性二阶偏微分方程方程的最佳控制问题，并具有分段平滑但在领导期间没有可观的非线性。关键参数取决于对国家级别集的分析。具体而言，我们表明，如果一个函数在边界上消失，并且其梯度与级别集的零不同，则该集合将分解为有限的许多封闭的简单曲线。此外，级别集合不断取决于定义这些集合的函数。我们还证明了级别集合积分的连续性。特别是，格林的第一个身份被证明适用于由两个具有非变化梯度的函数确定的开放集。在本文的第二部分中，明确的二阶条件将用于得出控制问题的有限元离散化的错误估计。
In this paper, we derive explicit second-order necessary and sufficient optimality conditions of a local minimizer to an optimal control problem for a quasilinear second-order partial differential equation with a piecewise smooth but not differentiable nonlinearity in the leading term. The key argument rests on the analysis of level sets of the state. Specifically, we show that if a function vanishes on the boundary and its the gradient is different from zero on a level set, then this set decomposes into finitely many closed simple curves. Moreover, the level sets depend continuously on the functions defining these sets. We also prove the continuity of the integrals on the level sets. In particular, Green's first identity is shown to be applicable on an open set determined by two functions with nonvanishing gradients. In the second part to this paper, the explicit sufficient second-order conditions will be used to derive error estimates for a finite-element discretization of the control problem.
