论文标题
最小二乘有限元素，用于分布式最佳控制问题
Least-squares finite elements for distributed optimal control problems
论文作者
论文摘要
我们为基于最小二乘有限元方法的分布式最佳控制问题的数值近似提供了一个框架。我们提出的方法同时解决了状态和伴随方程，并且是$ \ inf $ - $ \ sup $ stable，用于任何符合离散空间的选择。可靠有效的后验误差估计器是针对对控件施加框约束的问题的。它可以定位，因此用于引导自适应算法。对于不受限制的最佳控制问题，即，控制集合是希尔伯特空间，我们获得了一种强制性最小二乘法，尤其是用于任何选择离散近似空间的准典型性。对于受限的问题，我们得出并分析了最小二乘有限元方法解决PDE部分的变异不平等。我们表明，抽象框架可以应用于广泛的问题，包括标量二阶PDE，Stokes问题和时空域上的抛物线问题。提出了一些选定问题的数值示例。
We provide a framework for the numerical approximation of distributed optimal control problems, based on least-squares finite element methods. Our proposed method simultaneously solves the state and adjoint equations and is $\inf$--$\sup$ stable for any choice of conforming discretization spaces. A reliable and efficient a posteriori error estimator is derived for problems where box constraints are imposed on the control. It can be localized and therefore used to steer an adaptive algorithm. For unconstrained optimal control problems, i.e., the set of controls being a Hilbert space, we obtain a coercive least-squares method and, in particular, quasi-optimality for any choice of discrete approximation space. For constrained problems we derive and analyze a variational inequality where the PDE part is tackled by least-squares finite element methods. We show that the abstract framework can be applied to a wide range of problems, including scalar second-order PDEs, the Stokes problem, and parabolic problems on space-time domains. Numerical examples for some selected problems are presented.
