反常扩散方程状态受限最优控制问题的谱方法  

A spectral method for optimal control problems governed by the abnormal diffusion equation with integral constraint on state

作　　者：叶星旸[1] 许传炬[2,3] 

机构地区：[1]集美大学理学院,厦门361021 [2]厦门大学数学科学学院,厦门361005 [3]福建省数学建模与高性能科学计算重点实验室,厦门361005

出　　处：《中国科学：数学》2016年第7期1053-1070,共18页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:91130002;11471274和11421110001)资助项目

摘　　要：本文研究状态变量积分受限的反常扩散方程最优控制问题的时空谱方法,其控制方程为一个时间分数阶扩散方程.本文利用最优化理论中的Kuhn-Tucker条件分别推导了连续和离散的最优控制问题的最优性条件,分析了谱离散解的先验误差估计,并利用梯度投影算法求解离散最优化问题.最后通过数值算例验证了理论分析结果.In this paper, we study the space-time spectral approximation to an optimal control problem governed by the time fractional diffusion equation with an integral constraint on the state variable. The optimality conditions of the exact and discrete optimal control systems are derived in virtue of the Kuhn-Tucker condition.Then some a priori error estimates showing the spectral accuracy are obtained. Furthermore, an iterative algorithm based on the gradient projection optimization method using Galerkin spectral approximation is proposed to solve the discrete system. Finally, some numerical examples are performed to verify the theoretical results.

关 键 词：分数阶扩散方程 最优控制 谱方法 先验误差 

分 类 号：O241.82[理学—计算数学]