Lectures on the calculus of variations and optimal control theory (2024)

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Lectures on the calculus of variations and optimal control theory (1)

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英语 [en], .djvu, 🚀/lgli/lgrs/zlib, 5.6MB, 📘 非小说类图书, lgli/M_Mathematics/MOc_Optimization and control/Young L.C. Lectures on the calculus of variations and optimal control theory (2ed., Chelsea, 1980)(ISBN 082840304X)(600dpi)(K)(T)(O)(348s)_MOc_.djvu

Lectures on the calculus of variations and optimal control theory 🔍

Chelsea, 2., (corr.) ed, 1980

Laurence C Young 🔍

描述

Title......Page 1
Foreword......Page 3
Preface......Page 5
Contents......Page 7
Volume 1- Lectures on the Calculus of Variations......Page 12
1. Introduction......Page 13
2. The Place of the Calculus of Variations in Relation to the Rest of Mathematics and to Space Science......Page 14
3. Statement of the Simplest Problem and Some Cognate Matters......Page 15
4. Extremals in Some Classical Problems......Page 18
5. Solutions of the Preceding Problems (a), (b), (c)......Page 20
6. The Euler-Lagrange Lemma and Schwartz Distributions......Page 27
7. Alternative Forms of the Lemma......Page 29
8. Proof of the Main Form of the Lemma......Page 30
9. First Variation, Euler Equation, Transversality......Page 32
10. Perron's Paradox......Page 33
11. Introduction......Page 35
12. The Variational Algorithm ofHuygens......Page 36
13. A Link with Elementary Convexity......Page 38
14. Reappearance of the Euler Equation......Page 41
15. The Theorem of Malus......Page 43
16. Sufficient Conditions for Independence of the Hilbert Integral......Page 45
17. lnvariance Properties and an Envelope Theorem......Page 46
18. General Comments and the Applications to Plane Problems......Page 49
19. Background on Fix-Points and on Existence Theorems for Differential Equations and Implicit Functions......Page 51
21. The Legendre Transformation......Page 57
22. The Hamiltonian and its Properties......Page 58
23. Cauchy Characteristics......Page 60
24. Duality and the Standard Hamiltonian in the Parametric Case......Page 61
25. Other Admissible Parametric Hamiltonians......Page 64
26. Local Passage from Parametric to Nonparametric Case......Page 66
27. The Embedding of Small Extremals in Small Tubes......Page 67
28. Local Existence Theory for Nonparametric Variational Problems and for Ordinary Second Order Differential Equations......Page 69
29. Local Parametric Existence Theory for the Elliptic Case......Page 75
30. Introduction......Page 81
31. First and Second Variations and Transversality......Page 82
32. The Second Variation Fallacy......Page 84
33. The Secondary Hamiltonian......Page 85
34. Geometrical Interpretation of Exactness......Page 87
35. Distinguished Families......Page 89
36. Canonical Embeddings and Focal Points......Page 92
37. The Jacobi Theory of Conjugate Points......Page 94
38. The Index of Stability of an Extremal......Page 99
39. The Second Stage of the Morse Theory......Page 103
40. Introduction......Page 105
41. Center of Gravity and Dispersal Zone......Page 106
42. Convexity and the Hahn-Banach Theorem......Page 109
43. The Conceptual Heritage of Georg Cantor......Page 112
44. Duality of Convex Figures......Page 116
45. Duality of Convex Functions......Page 119
46. Hamiltonians in the Large and Reformulated Variational Theory......Page 121
47. Remarks on Classical Inequalities......Page 123
48. The Dual Unit Ball of a Functional Space......Page 124
49. The Riesz Representation......Page 129
50. Introduction......Page 133
51. The Hilbert Construction and Some of its Consequences in the Standard Parametric Case......Page 134
52. The Parametric Theory of Conjugate Points and the Parametric Jacobi Condition......Page 139
53. The Tonelli-Caratheodory Unicity Theorem......Page 144
54. Absolute and hom*otopic Minima on B · · i-Compact Domains and Manifolds......Page 154
55. Toward an Automatic Existence Theory......Page 158
56. First Stage of an Abstract Approach: Semicontinuity in a B · · i- Compact Set......Page 162
57, 58, 59......Page 165
60. Introduction......Page 166
61. Intuitive Background......Page 167
62. A Question of Semantics......Page 171
63. Parametric Curves in the Calculus of Variations......Page 172
64. Admissible Curves as Elements of a Dual Space......Page 174
65. A Human Analogy......Page 176
66. Generalized Curves and Flows, and Their Boundaries......Page 177
67. Parametric Representation of Generalized Curves......Page 182
68. Existence of a Minimum......Page 189
69. The Nature of the Generalized Solutions......Page 190
71. Separation Theorem for a Convex Cone in '1&'0 (A)......Page 195
72. The Lemma of the Insufficient Radius......Page 196
73. The Dual Separation Theorem......Page 198
74. A Localization Lemma for a B · · i-Compact Set......Page 199
75. Riesz Measures......Page 200
77. An Elementary Norm Estimate......Page 201
78. Vector Integration......Page 202
79. Closure of a Convex Hull......Page 203
80. Introduction......Page 205
81. Polygonal Flows......Page 206
83. The Variational Convexity Principle in its Elementary Form......Page 208
84. A First Extension......Page 209
85. The Enlargement Principle and the First Closure Theorem for Generalized Flows......Page 210
86. The Extension to Consistent Flows and Boundaries......Page 211
87. Preliminary Information on Mixtures and on the Lagrange Representation......Page 213
88. Further Comments on Measures, Mixtures, and Consistent Flows......Page 215
89. The Lagrange Representation of a Consistent Flow......Page 220
Volume II - Optimal Control

元数据中的注释

Bibliography: p. 325-329.
Includes index.

元数据中的注释

Kolxo3 -- 63-64

元数据中的注释

lg925410

替代作者

Laurence Chisholm. Young

替代作者

L. C. Young

备用出版社

New York : Chelsea Pub. Co., 1980.

备用出版社

Chelsea Publ. Co

备用版本

2d (corr.) ed., New York, New York (State), 1980

备用版本

2., (corr.) ed, New York, NY, 1980

备用版本

2nd ed. corr, New York, 1980

替代描述

Divided into two parts, this title first addresses the simpler variational problems in parametric and nonparametric form. It then covers extensions to optimal control theory. It notes that originally these problems were formulated as problems of Lagrange and Mayer in terms of differential constraints.

备用文件名

lgrsnf/M_Mathematics/MOc_Optimization and control/Young L.C. Lectures on the calculus of variations and optimal control theory (2ed., Chelsea, 1980)(ISBN 082840304X)(600dpi)(K)(T)(O)(348s)_MOc_.djvu

开源日期

2013-05-29T16:00:00

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Lectures on the calculus of variations and optimal control theory (2024)

FAQs

Lectures on the calculus of variations and optimal control theory? ›

In the optimal control formulation, such constraints are incorporated very naturally by working with an appropriate control set. In calculus of variations, on the other hand, they would make the description of the space of admissible curves quite cumbersome.

What is the difference between optimal control and calculus of variations? ›

In the optimal control formulation, such constraints are incorporated very naturally by working with an appropriate control set. In calculus of variations, on the other hand, they would make the description of the space of admissible curves quite cumbersome.

What is the calculus of variations and control theory? ›

The calculus of variations is used to find functions that optimize quantities expressed in terms of integrals. Optimal control theory seeks to find functions that minimize cost integrals for systems described by differential equations.

What is the calculus of variation theory? ›

The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers.

Who is the father of calculus of variations? ›

The calculus of variations may be said to begin with a problem of Johann Bernoulli (1696). It immediately occupied the attention of Jakob Bernoulli but Leonhard Euler first elaborated the subject. His contributions began in 1733, and his Elementa Calculi Variationum gave to the science its name.

Is calculus of variations useful? ›

It is also a powerful mathematical tool, finding applications in subjects as diverse as statics, optics, differential geometry, approximate solutions of differential equations and control theory.

What is the principle of calculus of variations? ›

A typical problem in the calculus of variations involve finding a particular function y(x) to maximize or minimize the integral I(y) subject to boundary conditions y(a)=A and y(b)=B. The integral I(y) is an example of a functional, which (more generally) is a mapping from a set of allowable functions to the reals.

What is the calculus of variations for beginners? ›

In calculus of variations the basic problem is to find a function y for which the functional I(y) is maximum or minimum. We call such functions as extremizing functions and the value of the functional at the extremizing function as extremum.

What is the fundamental concept behind the calculus of variation? ›

In mathematics, specifically in the calculus of variations, a variation δf of a function f can be concentrated on an arbitrarily small interval, but not a single point.

What are the three types of control theory? ›

Nonlinear, multivariable, adaptive and robust control theories come under this division.

What is an example of calculus of variations? ›

Calculus of variations is the area of mathematics concerned with optimizing mathematical objects called functionals. Calculus of variations can be used, for example, to find the shortest path on a surface or in physics, to describe the motion of a relativistic particle under gravity.

What is the prerequisite for calculus of variations? ›

The only prerequisites are several variable calculus and the rudiments of linear algebra and differential equations.

Is calculus of variations part of functional analysis? ›

Functional analysis owes much of its early impetus to problems that arise in the calculus of variations. In turn, the methods developed there have been applied to optimal control, an area that also requires new tools, such as nonsmooth analysis. This self-contained textbook gives a complete course on all these topics.

What are the techniques of the calculus of variation? ›

Their basic idea was to set up an integral for the total time of fall in terms of the unknown curve and then vary the curve so that a minimum time is obtained. This technique, typical of the calculus of variations, led to a differential equation whose solution is a curve called the cycloid.

What is the application of calculus of variations? ›

Calculus of variations help to formulate Geodesic problems on a plane and sphere. There are many laws of Physics which are written as variational principles. The Principle of Least action is equivalent to Newton Second Law of motion in a mechanical system.

What is the difference between control and optimal control? ›

A control problem includes a cost functional that is a function of state and control variables. An optimal control is a set of differential equations describing the paths of the control variables that minimize the cost function.

What is the difference between control and variation? ›

Control refers to the original version of your website where no change is made. The control is the base against which all your website test results are compared. Variation is the modified version of the website that you want to test against Control.

What is the difference between optimal control and model predictive control? ›

Optimal control generally refers to open-loop control, while MPC is closed-loop control: In both methods you use a model to predict the system behaviour. In optimal control you compute a sequence of input signals that steer this system.

What is the difference between calculus of variations and functional analysis? ›

Calculus of variations is a tool used to solve a specific set of problems. It started with brachistochrone problem. Functional analysis is a formulation of a general set of concepts attached to functions. To take an example, a metric on the set of functions is a concept formulated in functional analysis.

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