论文标题
无差距二阶最优条件，用于最佳控制非平滑偏椭圆形方程
No-gap second-order optimality conditions for optimal control of a non-smooth quasilinear elliptic equation
论文作者
论文摘要
本文介绍了二阶最佳条件，用于准线性椭圆控制问题，其主要部分的非线性系数为$ pc^2 $（连续和$ c^2 $，但与数量相比，$ c^2 $相距甚远）。我们证明，即使非线性系数不平滑，控制对状态运算符也是连续可区分的。这使我们能够在抽象曲率功能上建立“无隙”二阶必需和足够的最佳条件。例如，对于不平等是严格的事实，足够的条件与必要的条件有所不同。相当于二阶足够最佳条件的条件，也可用于误差估计，例如，还提供了有限元离散化。
This paper deals with second-order optimality conditions for a quasilinear elliptic control problem with a nonlinear coefficient in the principal part that is countably $PC^2$ (continuous and $C^2$ apart from countably many points). We prove that the control-to-state operator is continuously differentiable even though the nonlinear coefficient is non-smooth. This enables us to establish "no-gap" second-order necessary and sufficient optimality conditions in terms of an abstract curvature functional, i. e., for which the sufficient condition only differs from the necessary one in the fact that the inequality is strict. A condition that is equivalent to the second-order sufficient optimality condition and could be useful for error estimates in, e.g., finite element discretizations is also provided.
