- Артикул:00-01120751
- Автор: Richard Vinter
- ISBN: 978-0-8176-4990-6
- Обложка: Твердая обложка
- Издательство: Springer (все книги издательства)
- Город: Boston
- Страниц: 528
- Год: 2000
- Вес: 1469 г
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Издание на английском языке
This book is a comprehensive textbook on optimal control and variational analysis. It covers in detail fundamental principles and methods, such as optimal control theory, the calculus of variations, the existence of solutions, and the theory of hyperbolic equations. Particular attention is paid to the analysis of unsmoothed functions, subgradient costing, and necessary optimality conditions, such as maximum principles and extended Euler and Hamilton conditions. The book also covers topics such as differential inclusions, variational principles, constraint conditions, and the regularity of minimizing solutions. A significant portion is devoted to dynamic programming, the analysis of complex systems with states and control, and issues of safety and constraints in control systems.
Content
Preface
Notation
1. Overview
1.1. Optimal Control
1.2. The Calculus of Variations
1.3. Existence of Minimizers and Tonelli's Direct Method
1.4. Sufficient Conditions and the Hamilton-Jacobi Equation
1.5. The Maximum Principle
1.6. Dynamic Programming
1.7. Nonsmoothness
1.8. Nonsmooth Analysis
1.9. Nonsmooth Optimal Control
1.10. Epilogue
1.11. Notes for Chapter 1
2. Measurable Multifunctions and Differential Inclusions
2.1. Introduction
2.2. Convergence of Sets
2.3. Measurable Multifunctions
2.4. Existence and Estimation of F-Trajectories
2.5. Perturbed Differential Inclusions
2.6. Existence of Minimizing F-Trajectories
2.7. Relaxation
2.8. The Generalized Bolza Problem
2.9. Notes for Chapter 2
3. Variational Principles
3.1. Introduction
3.2. Exact Penalization
3.3. Ekeland's Theorem
3.4. Mini-Max Theorems
3.5. Notes for Chapter 3
4. Nonsmooth Analysis
4.1. Introduction
4.2. Normal Cones
4.3. Subdifferentials
4.4. Difference Quotient Representations
4.5. Nonsmooth Mean Value Inequalities
4.6. Characterization of Limiting Subgradients
4.7. Subgradients of Lipschitz Continuous Functions
4.8. The Distance Function
4.9. Criteria for Lipschitz Continuity
4.10. Relationships Between Normal and Tangent Cones
5. Subdifferential Calculus
5.1. Introduction
5.2. A Marginal Function Principle
5.3. Partial Limiting Subgradients
5.4. A Sum Rule
5.5. A Nonsmooth Chain Rule
5.6. Lagrange Multiplier Rules
5.7. Notes for Chapters 4 and 5
6. The Maximum Principle
6.1. Introduction
6.2. The Maximum Principle
6.3. Derivation of the Maximum Principle from the Extended Euler Condition
6.4. A Smooth Maximum Principle
6.5. Notes for Chapter 6
7. The Extended Euler-Lagrange and Hamilton Conditions
7.1. Introduction
7.2. Properties of the Distance Function
7.3. Necessary Conditions for a Finite Lagrangian Problem
7.4. The Extended Euler-Lagrange Condition: Nonconvex Velocity Sets
7.5. The Extended Euler-Lagrange Condition: Convex Velocity Sets
7.6. Dualization of the Extended Euler-Lagrange Condition
7.7. The Extended Hamilton Condition
7.8. Notes for Chapter 7
8. Necessary Conditions for Free End-Time Problems
8.1. Introduction
8.2. Lipschitz Time Dependence
8.3. Essential Values
8.4. Measurable Time Dependence
8.5. Proof of Theorem 8.4.1
8.6. Proof of Theorem 8.4.2
8.7. A Free End-Time Maximum Principle
8.8. Notes for Chapter 8
9. The Maximum Principle for State Constrained Problems
9.1. Introduction
9.2. Convergence of Measures
9.3. The Maximum Principle for Problems with State Constraints
9.4. Derivation of the Maximum Principle for State Constrained Problems from the Euler-Lagrange Condition
9.5. A Smooth Maximum Principle for State Constrained Problems
9.6. Notes for Chapter 9
10. Differential Inclusions with State Constraints
10.1. Introduction
10.2. A Finite Lagrangian Problem
10.3. The Extended Euler-Lagrange Condition for State Constrained Problems: Nonconvex Velocity Sets
10.4. Necessary Conditions for State Constrained Problems: Convex Velocity Sets
10.5. Free Time Problems with State Constraints
10.6. Nondegenerate Necessary Conditions
10.7. Notes for Chapter 10
11. Regularity of Minimizers
11.1. Introduction
11.2. Tonelli Regularity
11.3. Proof of The Generalized Tonelli Regularity Theorem
11.4. Lipschitz Continuous Minimizers
11.5. Autonomous Variational Problems with State Constraints
11.6. Bounded Controls
11.7. Lipschitz Continuous Controls
11.8. Notes for Chapter 11
12. Dynamic Programming
12.1. Introduction
12.2. Invariance Theorems
12.3. The Value Function and Generalized Solutions of the Hamilton-Jacobi Equation
12.4. Local Verification Theorems
12.5. Adjoint Arcs and Gradients of the Value Function
12.6. State Constrained Problems
12.7. Notes for Chapter 12
References
Index