Dynamic Programming and the Calculus of Variations

Name: Dynamic Programming and the Calculus of Variations
Brand: Academic Press
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Dreyfus(Author)

Dreyfus(Editor)

Academic Press

Published on 14. May 2014

271 pages

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978-0-08-095527-8 (ISBN)

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Front Cover
Dynamic Programming and the Calculus of Variations
Copyright Page
Contents
Preface
Chapter I. Discrete Dynamic Programming
1. Introduction
2. An Example of a Multistage Decision Process Problem
3. The Dynamic Programming solution of the Example
4. The Dynamic Programming Formalism
5. Two Properties of the Optimal Value Function
6. An Alternative Method of Solution
7. Modified Properties of the Optimal Value Function
8. A Property of Multistage Decision Processes
9. Further Illustrative Examples
10. Terminal Control Problems
11. Example of a Terminal Control Problem
12. Solution of the Example
13. Properties of the Solution of a Terminal Control Problem
14. Summary
Chapter II. The Classical Variational Theory
1. Introduction
2. A Problem
3. Admissible Solutions
4. Functions
5. Functionals
6. Minimization and Maximization
7. Arc-Length
8. The Simplest General Problem
9. The Maximum-Value Functional
10. The Nature of Necessary Conditions
11. Example
12. The Nature of Sufficent Conditions
13. Necessary and Sufficient Conditions
14. The Absolute Minimum of a Functional
15. A Relative Minimum of a Function
16. A Strong Relative Minimum of a Functional
17. A Weak Relative Minimum of a Functional
18. Weak Variations
19. The First and Second Variations
20. The Euler-Lagrange Equation
21. Example
22. The Legendre Condition
23. The Second Variation and the Second Derivative
24. The Jacobi Necessary Condition
25. Example
26. Focal Point
27. Geometric Conjugate Points
28. The Weierstrass Necessary Condition
29. Example
30. Discussion
31. Transversality Conditions
32. Corner Conditions
33. Relative Summary
34. Sufficient Conditions
35. Hamilton-Jacobi Theory
36. Other Problem Formulations
37. Example of a Terminal Control Problem
38. Necessary Conditions for the Problem of Mayer
39. Analysis of the Example Problem
40. Two-Point Boundary Value Problems
41. A Well-Posed Problem
42. Discussion
43. Computational Solution
44. Summary
References to Standard Texts
Chapter III. The Simplest Problem
1. Introduction
2. Notation
3. The Fundamental Partial Differential Equation
4. A Connection with Classical Variations
5. A Partial Differential Equation of the Classical Type
6. Two Kinds of Derivatives
7. Discussion of the Fundamental Partial Differential Equation
8. Characterization of the Optimal Policy Function
9. Partial Derivatives along Optimal Curves
10. Boundary Conditions for the Fundamental Equation: I
11. Boundary Conditions: II
12. An Illustrative Example-Variable End Point
13. A Further Example-Fixed Terminal Point
14. A Higher-Dimensional Example
15. A Different Method of Analytic Solution
16. An Example
17. From Partial to Ordinary Differential Equations
18. The Euler-Lagrange Equation
19. A Second Derivation of the Euler-Lagrange Equation
20. The Legendre Necessary Condition
21. The Weierstrass Necessary Condition
22. The Jacobi Necessary Condition
23. Discussion of the Jacobi Condition
24. Neighboring Optimal Solutions
25. An Illustrative Example
26. Determination of Focal Points
27. Example
28. Discussion of the Example
29. Transversality Conditions
30. Second-Order Transversality Conditions
31. Example
32. The Weierstrass Erdmann Corner Conditions
33. Summary
34. A Rigorus Dynamic Programming Approach
35. An Isoperimetric Problem
36. The Hamilton-Jacobi Equation
Chapter IV. The Problem of Mayer
1. Introduction
2. Statement of the Problem
3. The Optimal Value and Policy Functions
4. The Fundamental Partial Differential Equation
5. A Connection with the Simplest Problem
6. Interpretation of the Fundamental Equation
7. Boundary Conditions for the Fundamental Equation
8. Discussion
9. Two Necessary Conditions
10. The Multiplier Rule
11. The Clebsch Necessary Condition
12. The Weierstrass Necessary Condition
13. A Fourth Condition
14. The Second Partial Derivatives of the Optimal Value Function
15. A Matrix Differential Equation of Riccati Type
16. Terminal Values of the Second Partial Derivatives of the Optimal Value Function
17. The Guidance Problem
18. Terminal Transversality Conditions
19. Initial Transversality Conditions
20. Homogeneity
21. A First Integral of the Solution
22. The Variational Hamiltonian
23. Corner Conditions
24. An Example
25. Second-Order Transversality Conditions
26. Problem Discontinuities
27. Optimization of Parameters
28. A Caution
29. Summary
Chapter V. Inequality Constraints
1. Introduction
2. Control-Variable Inequality Constraints
3. The Appropriate Multiplier Rule
4. A Second Derivation of the Result of Section 3
5. Discussion
6. The Sign of the Control Impulse Response Function
7. Mixed Control-State Inequality Constraints
8. The Appropriate Modification of the Multiplier Rule
9. The Conventional Notation
10. A Second Derivation of the Result of Section 9
11. Discussion
12. The Sign of the New Multipier Function
13. State-Variable Inequality Constraints
14. The Optimal Value Function for a State-Constrained Problem
15. Derivation of a Multiplier Rule
16. Generalizations
17. A Connection with Other Forms of the Results
18. Summary
Chapter VI. Problems with Special Linear Structures
1. Introduction
2. Switching Manifolds
3. A Problem That Is Linear in the Derivative
4. Analysis of the Problem of Section 3
5. Discussion
6. A Problem with Linear Dynamics and Criterion
7. Investigation of the Problem of Section 6
8. Further Analysis of the Problem of Section 6
9. Discussion
10. Summary
Chapter VII. Stochastic and Adaptive Optimization Problems
1. Introduction
2. A Deterministic Problem
3. A Stochastic Problem
4. Discussion
5. Another Discrete Stochastic Problem
6. The Optimal Expected Value Function
7. The Fundamental Recurrence Relation
8. Discussion
9. A Continuous Stochastic Control Problem
10. The Optimal Expected Value Function
11. The Fundamental Partial Differential Equation
12. Discussion
13. The Analytic Solution of an Example
14. Discussion
15. A Modification of an Earlier Problem
16. Discussion
17. A Poisson Process
18. The Fundamental Partial Differential Equation for a Poisson Process
19. Adaptive Control Processes
20. A Numerical Problem
21. The Appropriate Prior-Probability Density
22. The State Variables
23. The Fundamental Recurrence Equation
24. A Further Specialization
25. Numerical Solution
26. Discussion
27. A Control Problem with Partially Observable States and with Deterministic Dynamics
28. Discussion
29. Sufficient Statistics
30. The Decomposition of Estimation and Control
31. A Warning
32. Summary
Bibliography
Author Index
Subject Index

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