中国人在弹性力学变分法的发明过程中也做出了重大贡献,弹性力学变分法准确地说叫做 "胡海昌- 鹫津久一郎"变分法。由胡海昌和鹫津久一郎相互独立地发明。

胡海昌. 弹性力学的变分原理及其应用. 北京：科学出版社，1981年5月第1版

作为数学的一个分支，变分法的诞生，是现实世界许多现象不断探索的结果，人们可以追寻到这样一个轨迹：

约翰·伯努利（Johann Bernoulli，1667－1748）1696年向全欧洲数学家挑战，提出一个难题：“设在垂直平面内有任意两点，一个质点受地心引力的作用，自较高点下滑至较低点，不计摩擦，问沿着什么曲线下滑，时间最短？”

这就是著名的“最速降线”问题（The Brachistochrone Problem）。它的难处在于和普通的极大极小值求法不同，它是要求出一个未知函数（曲线），来满足所给的条件。这问题的新颖和别出心裁引起了很大兴趣，罗比塔（Guillaume Francois Antonie de l'Hospital 1661-1704）、雅可比·伯努利（Jacob Bernoulli 1654-1705）、莱布尼茨（Gottfried Wilhelm Leibniz,1646-1716）和牛顿（Isaac Newton1642—1727）都得到了解答。约翰的解法比较漂亮，而雅可布的解法虽然麻烦与费劲，却更为一般化。后来欧拉（Euler Lonhard，1707～1783）和拉格朗日(Lagrange, Joseph Louis，1736-1813)发明了这一类问题的普遍解法，从而确立了数学的一个新分支——变分学。2100433B

《变分法》教学大纲

课程编码：07141001

课程名称：变分法

英文名称：Calculus of Variations

开课学期：第6学期

学时/学分：30/1.5 （其中实验学时：0学时）

课程类型：学位基础选修课

开课专业：机械科学与工程学院工程力学专业

选用教材：讲稿

主要参考书：《弹性力学》 徐芝纶编著高等教育出版社

《弹性和塑性力学的变分法》 鹫津久一郎著

《广义变分原理》钱伟长著

执笔人：周振平

变分法和最优控制论内容简介

《变分法和最优控制论(英文版)》适合数学、工程和相关专业的科研人员阅读。

变分法和最优控制论作者简介

作者：（美国）利伯逊（Daniel Liberzon）

Preface

1 Introduction

1.1 Optimal control problem

1.2 Some background on finite-dimensional optimization

1.2.1 Unconstrained optimization

1.2.2 Constrained optimization

1.3 Preview of infinite-dimensional optimization

1.3.1 Function spaces, norms, and local minima

1.3.2 First variation and first-order necessary condition.

1.3.3 Second variation and second-order conditions

1.3.4 Global minima and convex problems

1.4 Notes and references for Chapter 1

Calculus of Variations

2.1 Examples of variational problems

2.1.1 Dido's isoperimetric problem

2.1.2 Light reflection and refraction

2.1.3 Catenary

2.1.4 Brachistochrone

2.2 Basic calculus of variations problem

2.2.1 Weak and strong extrema

2.3 First-order necessary conditions for weak extrema

2.3.1 Euler-Lagrange equation

2.3.2 Historical remarks

2.3.3 Technical remarks

2.3.4 Two special cases

2.3.5 Variable-endpoint problems

2.4 Hamiltonian formalism and mechanics

2.4.1 Hamilton's canonical equations

2.4.2 Legendre transformation

2.4.3 Principle of least action and conservation laws

2.5 Variational problems with constraints

2.5.1 Integral constraints

2.5.2 Non-integral constraints

2.5 Second-order conditions

2.5.1 Legendre's necessary condition for a weak minimum

2.5.2 Sufficient condition for a weak minimum

2.7 Notes and references for Chapter 2

3 From Calculus of Variations to Optimal Control

3.1 Necessary conditions for strong extrema

3.1.1 Weierstrass-Erdmann corner conditions

3.1.2 Weierstrass excess function

3.2 Calculus of variations versus optimal control

3.3 Optimal control problem formulation and assumptions

3.3.1 Control system

3.3.2 Cost functional

3.3.3 Target set

3.4 Variational approach to the fixed-time, free-endpoint problem

3.4.1 Preliminaries

3.4.2 First variation

3.4.3 Second variation

3.4.4 Some comments

3.4.5 Critique of the variational approach and preview of the maximum principle

3.5 Notes and references for Chapter 3

The Maximum Principle

4.1 Statement of the maximum principle

4.1.1 Basic fixed-endpoint control problem

4.1.2 Basic variable-endpoint control problem

4.2 Proof of the maximum principle

4.2.1 From Lagrange to Mayer form

4.2.2 Temporal control perturbation

4.2.3 Spatial control perturbation

4.2.4 Variational equation

4.2.5 Terminal cone

4.2.5 Key topological lemma

4.2.7 Separating hyperplane

4.2.8 Adjoint equation

4.2.9 Properties of the Hamiltonian

4.2.10 Transversality condition

4.3 Discussion of the maximum principle

4.3.1 Changes of variables

4.4 Time-optimal control problems

4.4.1 Example: double integrator

4.4.2 Bang-bang principle for linear systems

4.4.3 Nonlinear systems, singular controls, and Lie brackets

4.4.4 Fuller's problem

4.5 Existence of optimal controls

4.5 Notes and references for Chapter 4

The Hamilton-Jacobi-Bellman Equation

5.1 Dynamic programming and the HJB equation

5.1.1 Motivation: the discrete problem

5.1.2 Principle of optimality

5.1.3 HJB equation

5.1.4 Sufficient condition for optimality

5.1.5 Historical remarks

5.2 HJB equation versus the maximum principle

5.2.1 Example: nondifferentiable value function

5.3 Viscosity solutions of the HJB equation

5.3.1 One-sided differentials

5.3.2 Viscosity solutions of PDEs

5.3.3 HJB equation and the value function

5.4 Notes and references for Chapter 5

6 The Linear Quadratic Regulator

6.1 Finite-horizon LQR problem

6.1.1 Candidate optimal feedback law

6.1.2 Riccati differential equation

6.1.3 Value function "and optimality

5.1.4 Global existence of solution for the RDE

5.2 Infinite-horizon LQR problem

6.2.1 Existence and properties of the limit

6.2.2 Infinite-horizon problem and its solution

5.2.3 Closed-loop stability

6.2.4 Complete result and discussion

6.3 Notes and references for Chapter 6

7 Advanced Topics

7.1 Maximum principle on manifolds

7.1.1 Differentiable manifolds

7.1.2 Re-interpreting the maximum principle

7.1.3 Symplectic geometry and Hamiltonian flows

7.2 HJB equation, canonical equations, and characteristics

7.2.1 Method of characteristics

7.2.2 Canonical equations as characteristics of the HJB equation

7.3 Piccati equations and inequalities in robust control

7.3.1 L2 gain

7.3.2 H∞ control problem

7.3.3 Riccati inequalities and LMIs

7.4 Maximum principle for hybrid control systems

7.4.1 Hybrid optimal control problem

7.4.2 Hybrid maximum principle

7.4.3 Example: light reflection

7.5 Notes and references for Chapter 7

Bibliography

Index 2100433B

我国山地分布广泛，边坡失稳引起的滑坡事故频发，造成水利、铁路、公路等设施严重破坏，同时给人民生命财产安全带来重大威胁。边坡稳定性问题是岩土力学与工程的基本问题之一，一直备受关注。实际边坡在破坏形式上呈现三维空间特性，然而传统的平面应变分析方法忽略了这种三维空间效应，不可避免会影响实际边坡稳定性评价，因此需要开展三维边坡稳定性分析。本项目将针对三维土质边坡稳定性极限分析研究中的不足，拟采用变分法构建三维破坏机制，通过理论分析和对比分析证明该三维机制的运动许可性和临界性，建立基于变分法的三维土质边坡稳定性极限分析方法，同时考虑水压力对三维边坡稳定性的影响，绘制一系列稳定图用于快速评价三维土质边坡稳定性，通过与平面应变解对比分析揭示三维空间效应对土质边坡稳定性的影响规律。研究成果将丰富边坡稳定性分析理论与评价方法，同时可以对实际边坡工程稳定性评估与加固也有较好的借鉴意义。