變分法

中國人在彈性力學(xué)變分法的發(fā)明過程中也做出了重大貢獻,彈性力學(xué)變分法準(zhǔn)確地說叫做 "胡海昌- 鷲津久一郎"變分法。由胡海昌和鷲津久一郎相互獨立地發(fā)明。

胡海昌. 彈性力學(xué)的變分原理及其應(yīng)用. 北京：科學(xué)出版社，1981年5月第1版

作為數(shù)學(xué)的一個分支，變分法的誕生，是現(xiàn)實世界許多現(xiàn)象不斷探索的結(jié)果，人們可以追尋到這樣一個軌跡：

約翰·伯努利（Johann Bernoulli，1667－1748）1696年向全歐洲數(shù)學(xué)家挑戰(zhàn)，提出一個難題：“設(shè)在垂直平面內(nèi)有任意兩點，一個質(zhì)點受地心引力的作用，自較高點下滑至較低點，不計摩擦，問沿著什么曲線下滑，時間最短？”

這就是著名的“最速降線”問題（The Brachistochrone Problem）。它的難處在于和普通的極大極小值求法不同，它是要求出一個未知函數(shù)（曲線），來滿足所給的條件。這問題的新穎和別出心裁引起了很大興趣，羅比塔（Guillaume Francois Antonie de l'Hospital 1661-1704）、雅可比·伯努利（Jacob Bernoulli 1654-1705）、萊布尼茨（Gottfried Wilhelm Leibniz,1646-1716）和牛頓（Isaac Newton1642—1727）都得到了解答。約翰的解法比較漂亮，而雅可布的解法雖然麻煩與費勁，卻更為一般化。后來歐拉（Euler Lonhard，1707～1783）和拉格朗日(Lagrange, Joseph Louis，1736-1813)發(fā)明了這一類問題的普遍解法，從而確立了數(shù)學(xué)的一個新分支——變分學(xué)。2100433B

《變分法》教學(xué)大綱

課程編碼：07141001

課程名稱：變分法

英文名稱：Calculus of Variations

開課學(xué)期：第6學(xué)期

學(xué)時/學(xué)分：30/1.5 （其中實驗學(xué)時：0學(xué)時）

課程類型：學(xué)位基礎(chǔ)選修課

開課專業(yè)：機械科學(xué)與工程學(xué)院工程力學(xué)專業(yè)

選用教材：講稿

主要參考書：《彈性力學(xué)》 徐芝綸編著高等教育出版社

《彈性和塑性力學(xué)的變分法》 鷲津久一郎著

《廣義變分原理》錢偉長著

執(zhí)筆人：周振平

變分法和最優(yōu)控制論內(nèi)容簡介

《變分法和最優(yōu)控制論(英文版)》適合數(shù)學(xué)、工程和相關(guān)專業(yè)的科研人員閱讀。

變分法和最優(yōu)控制論作者簡介

作者：（美國）利伯遜（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

我國山地分布廣泛，邊坡失穩(wěn)引起的滑坡事故頻發(fā)，造成水利、鐵路、公路等設(shè)施嚴(yán)重破壞，同時給人民生命財產(chǎn)安全帶來重大威脅。邊坡穩(wěn)定性問題是巖土力學(xué)與工程的基本問題之一，一直備受關(guān)注。實際邊坡在破壞形式上呈現(xiàn)三維空間特性，然而傳統(tǒng)的平面應(yīng)變分析方法忽略了這種三維空間效應(yīng)，不可避免會影響實際邊坡穩(wěn)定性評價，因此需要開展三維邊坡穩(wěn)定性分析。本項目將針對三維土質(zhì)邊坡穩(wěn)定性極限分析研究中的不足，擬采用變分法構(gòu)建三維破壞機制，通過理論分析和對比分析證明該三維機制的運動許可性和臨界性，建立基于變分法的三維土質(zhì)邊坡穩(wěn)定性極限分析方法，同時考慮水壓力對三維邊坡穩(wěn)定性的影響，繪制一系列穩(wěn)定圖用于快速評價三維土質(zhì)邊坡穩(wěn)定性，通過與平面應(yīng)變解對比分析揭示三維空間效應(yīng)對土質(zhì)邊坡穩(wěn)定性的影響規(guī)律。研究成果將豐富邊坡穩(wěn)定性分析理論與評價方法，同時可以對實際邊坡工程穩(wěn)定性評估與加固也有較好的借鑒意義。