Stochastic Galerkin Method for Optimal Control Problem Governed by Random Elliptic PDE with State Constraints
Stochastic Galerkin Method for Optimal Control Problem Governed by Random Elliptic PDE with State Constraints
复制标题
状态约束下随机椭圆偏微分方程最优控制问题的随机伽辽金法
DOI：
10.1007/s10915-018-0823-6
复制
发表时间：
2019
影响因子：
2.5
通讯作者：
Liu Wenbin
中科院分区：
文献类型：
--
作者：
Shen Wanfang;Ge Liang;Liu Wenbin
本文研究了一类系数为随机场的椭圆型偏微分方程最优控制问题的随机Galerkin逼近格式。最优控制最小化平均状态约束下的成本泛函的期望。首先将随机椭圆型偏微分方程用广义多项式混沌展开表示，得到参数化最优控制问题。利用次微分学中的斯莱特条件，在文献中首次得到了状态约束随机最优控制问题的最优性充要条件.然后，我们建立了一个随机Galerkin格式来逼近空间空间和概率空间中的最优性系统。然后推导出状态变量、协状态变量和控制变量的先验误差估计。提出并分析了一种投影算法。数值例子来说明我们的理论结果。
In this paper, we investigate a stochastic Galerkin approximation scheme for an optimal control problem governed by an elliptic PDE with random field in its coefficients. The optimal control minimizes the expectation of a cost functional with mean-state constraints. We first represent the stochastic elliptic PDE in terms of the generalized polynomial chaos expansion and obtain the parameterized optimal control problems. By applying the Slater condition in the subdifferential calculus, we obtain the necessary and sufficient optimality conditions for the state-constrained stochastic optimal control problem for the first time in the literature. We then establish a stochastic Galerkin scheme to approximate the optimality system in the spatial space and the probability space. Then the a priori error estimates are derived for the state, the co-state and the control variables. A projection algorithm is proposed and analyzed. Numerical examples are presented to illustrate our theoretical results.