论文标题
分析和近似Allen-CAHN方程的最佳控制问题
Analysis and approximations of an optimal control problem for the Allen-Cahn equation
论文作者
论文摘要
本文的范围是与Allen-Cahn方程相关的最佳控制问题的分析和近似。使用满足点控制控制约束的分布式控件，可以将跟踪功能最小化受allen-cahn方程。证明了必要和二阶的必要条件和足够条件。为了将控制与状态和伴随状态映射的近似值考虑，最低级别的不连续的Galerkin - 及时 - 方案。在适当限制时间和空间离散参数的最大尺寸下，$ k $，$ h $在描述了接口层的厚度的参数$ε$方面，A-priori估计值是通过$ 1/ε$在多面有上的常数证明的。与以前的不受控制的Allen-Cahn问题的作品不同，我们的方法不依赖于频谱估计的近似值的构造，因此，我们的估计值在最佳控制设置施加的低规律性假设下是有效的。在解决方案及其离散近似不满足均匀时空界限的情况下，这些估计值也是有效的。这些估计值和合适的本地化技术，通过二阶条件（请参阅\ cite {arada-casas-troltzsch_2002，casas-mateos-troltzsch_2005，casas-raymond_2006，casas-raymond_2006，casas-mateos-raymond_2007}），可以在局部差异之间进行差异，以弥补其范围，以便在距离之间进行差异，以弥补距离的差异，并依次估计该差异。变量及其离散近似
The scope of this paper is the analysis and approximation of an optimal control problem related to the Allen-Cahn equation. A tracking functional is minimized subject to the Allen-Cahn equation using distributed controls that satisfy point-wise control constraints. First and second order necessary and sufficient conditions are proved. The lowest order discontinuous Galerkin - in time - scheme is considered for the approximation of the control to state and adjoint state mappings. Under a suitable restriction on maximum size of the temporal and spatial discretization parameters $k$, $h$ respectively in terms of the parameter $ε$ that describes the thickness of the interface layer, a-priori estimates are proved with constants depending polynomially upon $1/ ε$. Unlike to previous works for the uncontrolled Allen-Cahn problem our approach does not rely on a construction of an approximation of the spectral estimate, and as a consequence our estimates are valid under low regularity assumptions imposed by the optimal control setting. These estimates are also valid in cases where the solution and its discrete approximation do not satisfy uniform space-time bounds independent of $ε$. These estimates and a suitable localization technique, via the second order condition (see \cite{Arada-Casas-Troltzsch_2002,Casas-Mateos-Troltzsch_2005,Casas-Raymond_2006,Casas-Mateos-Raymond_2007}), allows to prove error estimates for the difference between local optimal controls and their discrete approximation as well as between the associated state and adjoint state variables and their discrete approximations
