论文标题
通过Bloch波法在最佳控制中的高阶均质化
High-order homogenization in optimal control by the Bloch wave method
论文作者
论文摘要
本文研究了一个线性季度椭圆形最佳控制问题，其中成本功能和状态方程涉及高度振荡的周期系数$ a^\ varepsilon $。小参数$ \ varepsilon> 0 $表示周期性长度。我们提出了一个高阶有效控制问题，具有恒定系数，该系数提供了具有错误$ o（\ varepsilon^m）$的原始系数的近似值，其中$ m \ in \ mathbb {n} $与一个人一样大。我们的分析依赖于最佳解决方案的Bloch波扩展，并以两个步骤进行。在第一步中，我们在泰勒系列中扩展了最低的Bloch特征值，以获得高阶有效最佳控制问题。在第二步中，原始问题和有效问题分别根据Bloch和傅立叶变换重写。这允许通过相应的变异不等式直接比较最佳控制问题。
This article examines a linear-quadratic elliptic optimal control problem in which the cost functional and the state equation involve a highly oscillatory periodic coefficient $A^\varepsilon$. The small parameter $\varepsilon>0$ denotes the periodicity length. We propose a high-order effective control problem with constant coefficients that provides an approximation of the original one with error $O(\varepsilon^M)$, where $M\in\mathbb{N}$ is as large as one likes. Our analysis relies on a Bloch wave expansion of the optimal solution and is performed in two steps. In the first step, we expand the lowest Bloch eigenvalue in a Taylor series to obtain a high-order effective optimal control problem. In the second step, the original and the effective problem are rewritten in terms of the Bloch and the Fourier transform, respectively. This allows for a direct comparison of the optimal control problems via the corresponding variational inequalities.
