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E-raamat: Introduction to the Calculus of Variations and Control with Modern Applications

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  • Ilmumisaeg: 28-Aug-2013
  • Kirjastus: CRC Press Inc
  • Keel: eng
  • ISBN-13: 9781466571402
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Introduction to the Calculus of Variations and Control with Modern ApplicationsSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 28-Aug-2013
  • Kirjastus: CRC Press Inc
  • Keel: eng
  • ISBN-13: 9781466571402
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Introduction to the Calculus of Variations and Control with Modern Applications provides the fundamental background required to develop rigorous necessary conditions that are the starting points for theoretical and numerical approaches to modern variational calculus and control problems. The book also presents some classical sufficient conditions and discusses the importance of distinguishing between the necessary and sufficient conditions.





In the first part of the text, the author develops the calculus of variations and provides complete proofs of the main results. He explains how the ideas behind the proofs are essential to the development of modern optimization and control theory. Focusing on optimal control problems, the second part shows how optimal control is a natural extension of the classical calculus of variations to more complex problems.





By emphasizing the basic ideas and their mathematical development, this book gives you the foundation to use these mathematical tools to then tackle new problems. The text moves from simple to more complex problems, allowing you to see how the fundamental theory can be modified to address more difficult and advanced challenges. This approach helps you understand how to deal with future problems and applications in a realistic work environment.
Preface xi
Acknowledgments xvii
I Calculus of Variations
1(306)
1 Historical Notes on the Calculus of Variations
3(14)
1.1 Some Typical Problems
6(5)
1.1.1 Queen Dido's Problem
6(1)
1.1.2 The Brachistochrone Problem
7(1)
1.1.3 Shape Optimization
8(3)
1.2 Some Important Dates and People
11(6)
2 Introduction and Preliminaries
17(74)
2.1 Motivating Problems
17(9)
2.1.1 Problem 1: The Brachistochrone Problem
17(1)
2.1.2 Problem 2: The River Crossing Problem
18(2)
2.1.3 Problem 3: The Double Pendulum
20(1)
2.1.4 Problem 4: The Rocket Sled Problem
21(1)
2.1.5 Problem 5: Optimal Control in the Life Sciences
22(2)
2.1.6 Problem 6: Numerical Solutions of Boundary Value Problems
24(2)
2.2 Mathematical Background
26(31)
2.2.1 A Short Review and Some Notation
26(9)
2.2.2 A Review of One Dimensional Optimization
35(7)
2.2.3 Lagrange Multiplier Theorems
42(15)
2.3 Function Spaces
57(12)
2.3.1 Distances between Functions
64(4)
2.3.2 An Introduction to the First Variation
68(1)
2.4 Mathematical Formulation of Problems
69(17)
2.4.1 The Brachistochrone Problem
69(3)
2.4.2 The Minimal Surface of Revolution Problem
72(1)
2.4.3 The River Crossing Problem
73(1)
2.4.4 The Rocket Sled Problem
74(2)
2.4.5 The Finite Element Method
76(10)
2.5 Problem Set for
Chapter 2
86(5)
3 The Simplest Problem in the Calculus of Variations
91(40)
3.1 The Mathematical Formulation of the SPCV
91(4)
3.2 The Fundamental Lemma of the Calculus of Variations
95(7)
3.3 The First Necessary Condition for a Global Minimizer
102(15)
3.3.1 Examples
112(5)
3.4 Implications and Applications of the FLCV
117(8)
3.4.1 Weak and Generalized Derivatives
118(6)
3.4.2 Weak Solutions to Differential Equations
124(1)
3.5 Problem Set for
Chapter 3
125(6)
4 Necessary Conditions for Local Minima
131(54)
4.1 Weak and Strong Local Minimizers
132(3)
4.2 The Euler Necessary Condition - (I)
135(4)
4.3 The Legendre Necessary Condition - (III)
139(7)
4.4 Jacobi Necessary Condition - (IV)
146(9)
4.4.1 Proof of the Jacobi Necessary Condition
152(3)
4.5 Weierstrass Necessary Condition - (II)
155(23)
4.5.1 Proof of the Weierstrass Necessary Condition
159(12)
4.5.2 Weierstrass Necessary Condition for a Weak Local Minimum
171(5)
4.5.3 A Proof of Legendre's Necessary Condition
176(2)
4.6 Applying the Four Necessary Conditions
178(2)
4.7 Problem Set for
Chapter 4
180(5)
5 Sufficient Conditions for the Simplest Problem
185(18)
5.1 A Field of Extremals
186(4)
5.2 The Hilbert Integral
190(2)
5.3 Fundamental Sufficient Results
192(5)
5.4 Problem Set for
Chapter 5
197(6)
6 Summary for the Simplest Problem
203(10)
7 Extensions and Generalizations
213(70)
7.1 Properties of the First Variation
213(2)
7.2 The Free Endpoint Problem
215(9)
7.2.1 The Euler Necessary Condition
218(3)
7.2.2 Examples of Free Endpoint Problems
221(3)
7.3 The Simplest Point to Curve Problem
224(14)
7.4 Vector Formulations and Higher Order Problems
238(17)
7.4.1 Extensions of Some Basic Lemmas
242(7)
7.4.2 The Simplest Problem in Vector Form
249(3)
7.4.3 The Simplest Problem in Higher Order Form
252(3)
7.5 Problems with Constraints: Isoperimetric Problem
255(8)
7.5.1 Proof of the Lagrange Multiplier Theorem
259(4)
7.6 Problems with Constraints: Finite Constraints
263(2)
7.7 An Introduction to Abstract Optimization Problems
265(13)
7.7.1 The General Optimization Problem
265(2)
7.7.2 General Necessary Conditions
267(4)
7.7.3 Abstract Variations
271(2)
7.7.4 Application to the SPCV
273(1)
7.7.5 Variational Approach to Linear Quadratic Optimal Control
274(1)
7.7.6 An Abstract Sufficient Condition
275(3)
7.8 Problem Set for
Chapter 7
278(5)
8 Applications
283(24)
8.1 Solution of the Brachistochrone Problem
283(4)
8.2 Classical Mechanics and Hamilton's Principle
287(8)
8.2.1 Conservation of Energy
292(3)
8.3 A Finite Element Method for the Heat Equation
295(8)
8.4 Problem Set for
Chapter 8
303(4)
II Optimal Control
307(212)
9 Optimal Control Problems
309(32)
9.1 An Introduction to Optimal Control Problems
309(4)
9.2 The Rocket Sled Problem
313(2)
9.3 Problems in the Calculus of Variations
315(4)
9.3.1 The Simplest Problem in the Calculus of Variations
315(3)
9.3.2 Free End-Point Problem
318(1)
9.4 Time Optimal Control
319(19)
9.4.1 Time Optimal Control for the Rocket Sled Problem
319(14)
9.4.2 The Bushaw Problem
333(5)
9.5 Problem Set for
Chapter 9
338(3)
10 Simplest Problem in Optimal Control
341(32)
10.1 SPOC: Problem Formulation
341(2)
10.2 The Fundamental Maximum Principle
343(8)
10.3 Application of the Maximum Principle to Some Simple Problems
351(16)
10.3.1 The Bushaw Problem
351(7)
10.3.2 The Bushaw Problem: Special Case γ = 0 and κ = 1
358(4)
10.3.3 A Simple Scalar Optimal Control Problem
362(5)
10.4 Problem Set for
Chapter 10
367(6)
11 Extensions of the Maximum Principle
373(86)
11.1 A Fixed-Time Optimal Control Problem
373(4)
11.1.1 The Maximum Principle for Fixed t1
375(2)
11.2 Application to Problems in the Calculus of Variations
377(16)
11.2.1 The Simplest Problem in the Calculus of Variations
377(7)
11.2.2 Free End-Point Problems
384(1)
11.2.3 Point-to-Curve Problems
385(8)
11.3 Application to the Farmer's Allocation Problem
393(7)
11.4 Application to a Forced Oscillator Control Problem
400(4)
11.5 Application to the Linear Quadratic Control Problem
404(25)
11.5.1 Examples of LQ Optimal Control Problems
410(9)
11.5.2 The Time Independent Riccati Differential Equation
419(10)
11.6 The Maximum Principle for a Problem of Bolza
429(7)
11.7 The Maximum Principle for Nonautonomous Systems
436(10)
11.8 Application to the Nonautonomous LQ Control Problem
446(7)
11.9 Problem Set for
Chapter 11
453(6)
12 Linear Control Systems
459(60)
12.1 Introduction to Linear Control Systems
459(14)
12.2 Linear Control Systems Arising from Nonlinear Problems
473(5)
12.2.1 Linearized Systems
474(1)
12.2.2 Sensitivity Systems
475(3)
12.3 Linear Quadratic Optimal Control
478(2)
12.4 The Riccati Differential Equation for a Problem of Bolza
480(10)
12.5 Estimation and Observers
490(16)
12.5.1 The Luenberger Observer
494(4)
12.5.2 An Optimal Observer: The Kalman Filter
498(8)
12.6 The Time Invariant Infinite Interval Problem
506(3)
12.7 The Time Invariant Min-Max Controller
509(2)
12.8 Problem Set for
Chapter 12
511(8)
Bibliography 519(20)
Index 539
John Burns is the Hatcher Professor of Mathematics, Interdisciplinary Center for Applied Mathematics at Virginia Polytechnic Institute and State University. He is a fellow of the IEEE and SIAM. His research interests include distributed parameter control; approximation, control, identification, and optimization of functional and partial differential equations; aero-elastic control systems; fluid/structural control systems; smart materials; optimal design; and sensitivity analysis.
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