分布参数最优控制的边界元共轭梯度算法

李炳杰;刘三阳

系统科学与数学 ›› 2008, Vol. 28 ›› Issue (5) : 594-603.

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系统科学与数学 ›› 2008, Vol. 28 ›› Issue (5) : 594-603. DOI: 10.12341/jssms10231
论文

分布参数最优控制的边界元共轭梯度算法

    李炳杰(1), 刘三阳(2)
作者信息 +

Conjugate Gradient-Boundary Element Method to Distributed Optimal Control Problem

    LI Bingjie (1), LIU Sanyang(2)
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摘要

研究了一类线性椭圆型分布参数最优控制问题的数值解算法.得到最优控制对应的最优性方程组,
在凸性条件下, 证明了最优控制的唯一存在性问题.将最优控制问题化为以控制函数和状态函数为局中人的递阶式(Stackelberg)非合作对策问题,其平衡点是最优控制的解.进一步得到求平衡点的边界元共轭梯度算法.最后,研究算法中边界元离散的误差估计,以算例验证该算法.

Abstract

The numerical solution of distributed optimal control of a linear elliptic problem is investigated. The system of optimality consisting of state and costate variables (Lagrangian multiplier)for the optimal control is derived, and in convex condition, uniqueness of optimal solution is proved. The optimal control problem is translated into a kind of two players game problem which is a non-cooperative Stackelberg game between control function and state function.
The Nash equilibrium point for the new system is the solution of the optimal control problem. The conjugate gradient-boundary element method
for solving the Nash equilibrium point is developed. Finally, the
error estimates for these schemes are obtained. Numerical results
indicate that the approach can save substantial computational work and that the algorithm is effective.

关键词

分布参数最优控制 / 基本解 / 边界元方法 / Nash平衡点 / 共轭梯度算法.

Key words

Distributed optimal control / fundamental solution / boundary element method / Nash equilibrium point / conjugate gradient method.

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李炳杰 , 刘三阳. 分布参数最优控制的边界元共轭梯度算法. 系统科学与数学, 2008, 28(5): 594-603. https://doi.org/10.12341/jssms10231
LI Bingjie , LIU Sanyang. Conjugate Gradient-Boundary Element Method to Distributed Optimal Control Problem. Journal of Systems Science and Mathematical Sciences, 2008, 28(5): 594-603 https://doi.org/10.12341/jssms10231
中图分类号： 49J20    65N38    65N30    65N35   
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