求解最优控制问题的一类单调时间离散格式

向清清; 陈浩

doi:10.21656/1000-0887.460020

求解最优控制问题的一类单调时间离散格式

doi: 10.21656/1000-0887.460020

向清清,
陈浩^,

重庆师范大学数学科学学院, 重庆 401331

基金项目:

重庆市自然科学基金 CSTB2025NSCQ-GPX1015

详细信息

作者简介:
向清清(2000—)，女，硕士生(E-mail: 2530025502@qq.com)

通讯作者:
陈浩(1986—)，男，教授，博士，硕士生导师(通信作者. E-mail: hch@cqnu.edu.cn)

中图分类号: O241.81
计量
- 文章访问数: 13
- HTML全文浏览量: 12
- PDF下载量: 3
- 被引次数: 0
出版历程
- 收稿日期: 2025-02-06
- 修回日期: 2025-11-07
- 刊出日期: 2026-02-01

A Monotonic Time-Discretized Scheme for Optimal Control Problems

XIANG Qingqing,
CHEN Hao^,

College of Mathematics Science, Chongqing Normal University, Chongqing 401331, P.R.China

摘要

摘要: 近期, Breitenbach和Borzì构造了一类求解常微分方程最优控制问题的序列二次Hamilton(sequential quadratic Hamiltonian, SQH)方法. 他们证明了该迭代方法在连续时间情形下的单调收敛性. 然而, 该迭代方法在离散时间情形下的收敛性质尚未被解决. 该文构造了一类中点时间离散格式，并证明了其能保持SQH迭代的单调收敛性. 数值实验验证了该方法的有效性及收敛性.
- 最优控制问题 /
- 序列二次Hamilton方法 /
- 单调时间离散格式 /
- 收敛性
Abstract: Recently, Breitenbach and Borzì proposed a sequential quadratic Hamiltonian method for solving optimal control problems. They proved the monotonic convergence of the algorithm in the continuous time case. However, the properties of the discrete version of the iterative procedure have not been tackled yet. A midpoint time-discretized scheme preserving the monotonic properties of the sequential quadratic Hamiltonian method was presented. Numerical experiments show the effectiveness and convergence of the proposed algorithm.
- optimal control problem /
- sequential quadratic Hamiltonian method /
- monotonic time-discretized scheme /
- convergence

HTML全文

图 1 y₁-y₂平面上的运动轨迹

Figure 1. Motion paths in the y₁-y₂ plane

下载: 全尺寸图片幻灯片

图 2 目标函数J的收敛过程

Figure 2. The convergence history of J along the SQH iterations

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图 3 伴随变量

Figure 3. The adjoint variables

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图 4 最优控制

Figure 4. The optimal control

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图 5 目标泛函J的收敛过程

Figure 5. The convergence history of J along the SQH iterations

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表 1 NCG算法、SQH算法与Adam算法比较

Table 1. The comparison between the NCG algorithm, the SQH algorithm and the Adam algorithm

J	NCG		SQH		Adam
J	CPU time/s	iter	CPU time/s	iter	CPU time/s	iter
0.78	12.548 5	1 394	10.989 7	620	21.016 1	1 908
1.00	6.263 4	700	3.276 0	186	12.883 5	1 418
2.00	0.549 1	56	0.304 7	32	4.275 4	467
3.00	0.318 4	32	0.224 5	23	2.351 6	252
4.00	0.232 5	22	0.185 5	19	1.617 1	182
5.00	0.188 8	16	0.165 5	16	1.379 2	145

下载: 导出CSV

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