《变分法与最优控制和偏微分方程》（2023）

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课程简介

本课程主要讨论一下课题：

• 经典力学的基本数学理论：Hamilton系统与Lagrange系统
• 古典一维变分法（各类必要条件）
• 特征理论与Hamilton-Jacobi方程
• Tonelli存在性定理与部分正则性定理
• 一阶Hamilton-Jacobi方程粘性解理论、弱KAM理论
• 最优控制与Pontryagin极大原理

课程基础

本课程需要的基础为数学分析、线性代数、常微分方程、实变函数与泛函分析、偏微分方程

参考书目

• Kot, Mark A first course in the calculus of variations. Student Mathematical Library, 72. American Mathematical Society, 2014.
• Levi, Mark Classical mechanics with calculus of variations and optimal control. An intuitive introduction. Student Mathematical Library, 69. American Mathematical Society, 2014.
• Mesterton-Gibbons, Mike A primer on the calculus of variations and optimal control theory. Student Mathematical Library, 50. American Mathematical Society, 2009.
• Liberzon, Daniel Calculus of variations and optimal control theory. A concise introduction. Princeton University Press, 2012.
• Clarke, Francis Functional analysis, calculus of variations and optimal control. Graduate Texts in Mathematics, 264. Springer, 2013.
• Cannarsa, Piermarco; Sinestrari, Carlo Semiconcave functions, Hamilton-Jacobi equations, and optimal control. Progress in Nonlinear Differential Equations and their Applications, 58. Birkhäuser, 2004.
• Arnolʹd, V. I. Mathematical methods of classical mechanics. Second edition. Graduate Texts in Mathematics, 60. Springer-Verlag, 1989.
• Giaquinta, Mariano; Hildebrandt, Stefan Calculus of variations. I. The Lagrangian formalism. Grundlehren der Mathematischen Wissenschaften 310. Springer-Verlag, 1996.
• Giaquinta, Mariano; Hildebrandt, Stefan Calculus of variations. II. The Hamiltonian formalism. Grundlehren der Mathematischen Wissenschaften 311. Springer-Verlag, 1996.
• Buttazzo, Giuseppe; Giaquinta, Mariano; Hildebrandt, Stefan One-dimensional variational problems. An introduction. Oxford Lecture Series in Mathematics and its Applications, 15, Oxford University Press, 1998.
• Young, L. C. Lectures on the calculus of variations and optimal control theory, Second Edition, Chelsea Pub Co., 1969.
• Carathéodory, C. Calculus of Variations and Partial Differential Equations of First Order, Third Edition, American Mathematical Society, 1999.