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 the state to adjoint 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 Arada et al. in Comput Optim Appl 23(2):201–229, 2002; Casas et al. in Comput Optim Appl 31(2): 193–219, 2005; Casas and Raymond in SIAM J Control Optim 45(5):1586–1611, 2006; Casas et al. in Control Optim 46(3):952–982, 2007), allows to prove error estimates for the difference between local optimal controls and their discrete approximations as well as between the associated state and adjoint state variables and their discrete approximations.

本文的范围是与 Allen-Cahn 方程相关的最优控制问题的分析和近似。使用满足逐点控制约束的分布式控制，根据 Allen-Cahn 方程最小化跟踪函数。证明了一阶和二阶充要条件。时间上的最低阶不连续伽辽金方案被考虑用于控制到状态以及状态到伴随映射的近似。在对时间和空间离散化参数k、h的最大尺寸进行适当限制的情况下，分别以参数\(\epsilon \)表示描述界面层的厚度，先验估计通过多项式依赖于\(1/ \epsilon \) 的常数来证明。与之前针对不受控艾伦-卡恩问题的研究不同，我们的方法不依赖于谱估计近似的构造，因此，我们的估计在最优控制设置强加的低规律性假设下是有效的。这些估计在解及其离散近似不满足独立于\(\epsilon \)的均匀时空界限的情况下也是有效的。这些估计和合适的定位技术，通过二阶条件（参见 Arada 等人，Comput Optim Appl 23(2):201–229, 2002；Casas 等人，Comput Optim Appl 31(2): 193–219 ，2005 年；Casas 和 Raymond 在 SIAM J Control Optim 45(5):1586–1611, 2006 中；Casas 等人在 Control Optim 46(3):952–982, 2007 中），允许证明之间差异的误差估计局部最优控制及其离散近似，以及关联状态和伴随状态变量及其离散近似之间的关系。