摘 要：最优控制问题是现代控制理论中的核心课题，广泛应用于工程、经济及生物等领域。本文以变分法为基础，探讨其在解决最优控制问题中的理论与应用价值。研究旨在通过经典变分法的数学框架，结合庞特里亚金极小值原理和哈密尔顿函数方法，构建一套系统化的分析工具，用于求解具有约束条件的复杂最优控制问题。通过对典型控制模型的数值实验，验证了所提方法的有效性与精确性。研究发现，基于变分法的优化策略不仅能够显著提升计算效率，还能灵活处理多目标优化场景下的动态约束。此外，本文创新性地引入了改进型变分算法，有效克服了传统方法在高维问题中的局限性。结果表明，该方法在降低计算复杂度的同时，保持了较高的收敛精度。综上，本文为最优控制问题提供了一种高效且普适的解决方案，拓展了变分法在现代控制领域的应用边界，为后续研究奠定了坚实的理论基础。
关键词：最优控制；变分法；庞特里亚金极小值原理；改进型变分算法；多目标优化
An Exploration of the Application of Variational Methods in Optimal Control Problems
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Abstract：The optimal control problem is a central topic in modern control theory, with extensive applications in engineering, economics, and biology. Based on the calculus of variations, this study investigates its theoretical and practical significance in solving optimal control problems. The research aims to construct a systematic analytical toolkit by integrating the mathematical fr amework of classical variational methods with Pontryagin's Minimum Principle and the Hamiltonian function approach, specifically designed for complex optimal control problems with constraints. Numerical experiments conducted on typical control models validate the effectiveness and accuracy of the proposed methodology. It is found that optimization strategies grounded in variational methods not only significantly enhance computational efficiency but also demonstrate flexibility in handling dynamic constraints in multi-ob jective optimization scenarios. Additionally, an improved variational algorithm is innovatively introduced to effectively address the limitations of traditional methods in high-dimensional problems. Results indicate that this approach reduces computational complexity while maintaining high convergence precision. In summary, this study provides an efficient and universally applicable solution to optimal control problems, extending the application boundaries of variational methods in modern control theory and laying a solid theoretical foundation for future research.
Keywords: Optimal Control；Variational Method；Pontryagin Minimum Principle；Improved Variational Algorithm；Multi-ob jective Optimization
目 录
引言 1
一、变分法基础理论概述 1
(一)变分法的基本概念 1
(二)欧拉 2
(三)变分法在经典问题中的应用 2
二、最优控制问题的数学建模 3
(一)最优控制问题的定义与分类 3
(二)控制系统的状态方程构建 3
(三)性能指标函数的设计方法 4
三、变分法解决最优控制问题的核心方法 4
(一)哈密尔顿函数的引入与作用 5
(二)极值原理在变分法中的体现 5
(三)边界条件与约束处理技术 6
四、变分法在实际最优控制问题中的应用案例 6
(一)工程领域中的典型应用分析 6
(二)经济系统中的优化控制探索 2
(三)复杂动态系统的数值求解研究 2
结论 2
参考文献 4
致谢 4
