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 $\epsilon$ that describes the thickness of the interface layer, a-priori estimates are proved with constants depending polynomially upon $1/ \epsilon$. 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 $\epsilon$. 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

翻译：本文的范围是 Allen- Cahn 方程式相关最佳控制问题的分析和近似。 使用符合点控制限制的分布式控制,将跟踪功能在Allen- Cahn 方程式中最小化, 使用Allen- Cahn 方程式, 满足点控制限制。 第一和第二顺序必要和足够条件得到证明。 最低顺序不连续 Galerkin - 时间 - 考虑最低顺序的Galerkin - - 州和州联合绘图的近似控制。 根据对时间和空间离散参数最大尺寸的适当限制, 美元, 以显示界面厚度的参数差值为美元, 以显示Allen- Cahn 方程式的分布式控制为最小值, 以恒定常数为常数, 与未受控制的 Allen- Cal- Rahn 问题不同的是, 我们的方法并不依赖于光谱估计值的近似值, 因此, 在最佳控制环境所设定的低常规假设下, 我们的估算值是有效的。 这些估计在解决方案及其离子- Restal- C ral- Ral- sal- sal- sal- sal- salate- sal- surveal- supy- surview 和适当的当地技术, as- sild- sild- sild- sild- sild- sal- surviual- silds survic.