最优控制问题综述
A Review of Optimal Control Problems
摘要: 本文系统综述了偏微分方程(PDE)约束的最优控制问题。首先梳理了其从变分法到现代控制理论的发展历程，重点分析了“先离散后优化”与“先优化后离散”两类数值求解策略，探讨了有限差分法、有限元法等离散方法的特性。特别关注了带复杂约束的非光滑问题求解，评述了神经网络等新兴方法的进展。最后从算法创新、多物理场耦合等维度展望了未来发展方向，为研究者提供该领域的知识体系概览和发展趋势分析。
Abstract: This article systematically reviews the optimal control problems constrained by partial differential equations (PDEs). Firstly, it traces the development process from variational methods to modern control theory, focusing on analyzing the two numerical solution strategies of “discretization first and optimization second” and “optimization first and discretization second”. It also discusses the characteristics of discretization methods such as finite difference method and finite element method. Special attention is paid to the solution of non-smooth problems with complex constraints, and the progress of emerging methods such as neural networks is reviewed. Finally, it looks forward to future development directions from dimensions such as algorithm innovation and multi-physics field coupling, providing researchers with an overview of the knowledge system and trend analysis in this field.
文章引用：陈世奇. 最优控制问题综述[J]. 应用数学进展, 2026, 15(4): 362-370. https://doi.org/10.12677/aam.2026.154164

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