论文标题
双线性状态方程的最佳控制问题的后验错误估计
A posteriori error estimates for an optimal control problem with a bilinear state equation
论文作者
论文摘要
我们提出和分析了一个最佳控制问题的后验误差估计器，该估计值涉及一个椭圆形偏微分方程作为状态方程，而控制变量则以系数输入状态方程；也考虑了控制变量上的点约束。我们考虑两种不同的策略来近似最佳变量：一个完全离散的方案，其中可允许的控制集由分段常数函数离散和半混凝土方案，在该方案中，可允许的控制集未离散；后一种方案基于所谓的变分离散方法。我们为每种解决方案技术设计了一个后验误差估计器，并在两个和三维Lipschitz多边形/多面体结构域（不一定是凸）显示了所提出的误差估计器是可靠且有效的。我们根据设计的估计器设计的自适应策略设计，这些策略为执行的数值示例提供了最佳的实验速率。
We propose and analyze a posteriori error estimators for an optimal control problem that involves an elliptic partial differential equation as state equation and a control variable that enters the state equation as a coefficient; pointwise constraints on the control variable are considered as well. We consider two different strategies to approximate optimal variables: a fully discrete scheme in which the admissible control set is discretized with piecewise constant functions and a semi-discrete scheme where the admissible control set is not discretized; the latter scheme being based on the so-called variational discretization approach. We design, for each solution technique, an a posteriori error estimator and show, in two and three dimensional Lipschitz polygonal/polyhedral domains (not necessarily convex), that the proposed error estimator is reliable and efficient. We design, based on the devised estimators, adaptive strategies that deliver optimal experimental rates of convergence for the performed numerical examples.
