论文标题
一维椭圆形分布的最佳控制问题的有限元方法，对状态的衍生物有点约束
Finite Element Methods for One Dimensional Elliptic Distributed Optimal Control Problems with Pointwise Constraints on the Derivative of the State
论文作者
论文摘要
我们研究了$ c^1 $有限元方法，用于一维椭圆分布的最佳控制问题，对状态变量的第四阶变异不等式的派生派对状态的衍生物进行了重点约束。对于DIRICHLET边界条件的问题，我们使用现有的$ H^{\frac52-ε} $规则性结果，以使$ o（h^{\frac12-ε}）$收敛来获得$ h^2 $规范中最佳状态的近似值。对于混合Dirichlet和Neumann边界条件的问题，我们表明，在数据的适当假设下，最佳状态属于$ h^3 $，并获得$ o（h）$收敛以在$ h^2 $ norm中近似最佳状态。
We investigate $C^1$ finite element methods for one dimensional elliptic distributed optimal control problems with pointwise constraints on the derivative of the state formulated as fourth order variational inequalities for the state variable. For the problem with Dirichlet boundary conditions, we use an existing $H^{\frac52-ε}$ regularity result for the optimal state to derive $O(h^{\frac12-ε})$ convergence for the approximation of the optimal state in the $H^2$ norm. For the problem with mixed Dirichlet and Neumann boundary conditions, we show that the optimal state belongs to $H^3$ under appropriate assumptions on the data and obtain $O(h)$ convergence for the approximation of the optimal state in the $H^2$ norm.
