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E-raamat: Nonlinear Optimal Control Theory

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(North Carolina State University, Raleigh, USA), (Purdue University, West Lafayette, Indiana, USA)
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Nonlinear Optimal Control TheorySuurem pilt
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"Preface This book is an introduction to the mathematical theory of optimal control of processes governed by ordinary differential and certain types of differential equations with memory. The book is intended for students, mathematicians, and those who apply the techniques of optimal control in their research. Our intention is to give a broad, yet relatively deep, concise and coherent introduction to the subject. We have dedicated an entire chapter for examples. We have dealt with the examples pointing out the mathematical issues that one needs to address. The first six chapters can provide enough material for an introductory course in optimal control theory governed by differential equations. Chapters 3, 4, and 5 could be covered with more or less details in the mathematical issues depending on the mathematical background of the students. For students with background in functional analysis and measure theory Chapter 7 can be added. Chapter 7 is a more mathematically rigorous version of the material in Chapter 6. We have included material dealing with problems governed by integrodifferential and delay equations. We have given a unified treatment of bounded state problems governed by ordinary, integrodifferential, and delay systems. We have also added material dealing with the Hamilton-Jacobi Theory. This material sheds light on the mathematical details that accompany the material in Chapter 6"--



Nonlinear Optimal Control Theory presents a deep, wide-ranging introduction to the mathematical theory of the optimal control of processes governed by ordinary differential equations and certain types of differential equations with memory. Many examples illustrate the mathematical issues that need to be addressed when using optimal control techniques in diverse areas.

Drawing on classroom-tested material from Purdue University and North Carolina State University, the book gives a unified account of bounded state problems governed by ordinary, integrodifferential, and delay systems. It also discusses Hamilton-Jacobi theory. By providing a sufficient and rigorous treatment of finite dimensional control problems, the book equips readers with the foundation to deal with other types of control problems, such as those governed by stochastic differential equations, partial differential equations, and differential games.

Arvustused

This book provides a thorough introduction to optimal control theory for nonlinear systems. The book is enhanced by the inclusion of many examples, which are analyzed in detail using Pontryagins principle. An important feature of the book is its systematic use of a relaxed control formulation of optimal control problems. From the Foreword by Wendell Fleming

more than a very useful research account and a handy reference to users of the theory-they also make it a pleasant and helpful study opportunity to students and other newcomers to the theory of optimal control. Zvi Artstein, in Mathematical Reviews

Foreword ix
Preface xi
1 Examples of Control Problems
1(14)
1.1 Introduction
1(1)
1.2 A Problem of Production Planning
1(2)
1.3 Chemical Engineering
3(1)
1.4 Flight Mechanics
4(3)
1.5 Electrical Engineering
7(2)
1.6 The Brachistochrone Problem
9(3)
1.7 An Optimal Harvesting Problem
12(1)
1.8 Vibration of a Nonlinear Beam
13(2)
2 Formulation of Control Problems
15(20)
2.1 Introduction
15(1)
2.2 Formulation of Problems Governed by Ordinary Differential Equations
15(3)
2.3 Mathematical Formulation
18(4)
2.4 Equivalent Formulations
22(4)
2.5 Isoperimetric Problems and Parameter Optimization
26(1)
2.6 Relationship with the Calculus of Variations
27(5)
2.7 Hereditary Problems
32(3)
3 Relaxed Controls
35(44)
3.1 Introduction
35(3)
3.2 The Relaxed Problem; Compact Constraints
38(5)
3.3 Weak Compactness of Relaxed Controls
43(13)
3.4 Filippov's Lemma
56(7)
3.5 The Relaxed Problem; Non-Compact Constraints
63(3)
3.6 The Chattering Lemma; Approximation to Relaxed Controls
66(13)
4 Existence Theorems; Compact Constraints
79(34)
4.1 Introduction
79(1)
4.2 Non-Existence and Non-Uniqueness of Optimal Controls
80(3)
4.3 Existence of Relaxed Optimal Controls
83(9)
4.4 Existence of Ordinary Optimal Controls
92(6)
4.5 Classes of Ordinary Problems Having Solutions
98(3)
4.6 Inertial Controllers
101(2)
4.7 Systems Linear in the State Variable
103(10)
5 Existence Theorems; Non-Compact Constraints
113(36)
5.1 Introduction
113(1)
5.2 Properties of Set Valued Maps
114(3)
5.3 Facts from Analysis
117(5)
5.4 Existence via the Cesari Property
122(17)
5.5 Existence Without the Cesari Property
139(6)
5.6 Compact Constraints Revisited
145(4)
6 The Maximum Principle and Some of Its Applications
149(56)
6.1 Introduction
149(1)
6.2 A Dynamic Programming Derivation of the Maximum Principle
150(9)
6.3 Statement of Maximum Principle
159(14)
6.4 An Example
173(4)
6.5 Relationship with the Calculus of Variations
177(5)
6.6 Systems Linear in the State Variable
182(4)
6.7 Linear Systems
186(6)
6.8 The Linear Time Optimal Problem
192(1)
6.9 Linear Plant-Quadratic Criterion Problem
193(12)
7 Proof of the Maximum Principle
205(44)
7.1 Introduction
205(2)
7.2 Penalty Proof of Necessary Conditions in Finite Dimensions
207(3)
7.3 The Norm of a Relaxed Control; Compact Constraints
210(2)
7.4 Necessary Conditions for an Unconstrained Problem
212(6)
7.5 The ε-Problem
218(5)
7.6 The ε-Maximum Principle
223(5)
7.7 The Maximum Principle; Compact Constraints
228(6)
7.8 Proof of Theorem 6.3.9
234(4)
7.9 Proof of Theorem 6.3.12
238(2)
7.10 Proof of Theorem 6.3.17 and Corollary 6.3.19
240(4)
7.11 Proof of Theorem 6.3.22
244(5)
8 Examples
249(34)
8.1 Introduction
249(1)
8.2 The Rocket Car
249(6)
8.3 A Non-Linear Quadratic Example
255(2)
8.4 A Linear Problem with Non-Convex Constraints
257(2)
8.5 A Relaxed Problem
259(3)
8.6 The Brachistochrone Problem
262(5)
8.7 Flight Mechanics
267(6)
8.8 An Optimal Harvesting Problem
273(3)
8.9 Rotating Antenna Example
276(7)
9 Systems Governed by Integrodifferential Systems
283(12)
9.1 Introduction
283(1)
9.2 Problem Statement
283(2)
9.3 Systems Linear in the State Variable
285(2)
9.4 Linear Systems/The Bang-Bang Principle
287(1)
9.5 Systems Governed by Integrodifferential Systems
287(1)
9.6 Linear Plant Quadratic Cost Criterion
288(1)
9.7 A Minimum Principle
289(6)
10 Hereditary Systems
295(10)
10.1 Introduction
295(1)
10.2 Problem Statement and Assumptions
295(1)
10.3 Minimum Principle
296(2)
10.4 Some Linear Systems
298(2)
10.5 Linear Plant-Quadratic Cost
300(1)
10.6 Infinite Dimensional Setting
300(5)
10.6.1 Approximate Optimality Conditions
302(2)
10.6.2 Optimality Conditions
304(1)
11 Bounded State Problems
305(32)
11.1 Introduction
305(1)
11.2 Statement of the Problem
305(1)
11.3 ε-Optimality Conditions
306(10)
11.4 Limiting Operations
316(4)
11.5 The Bounded State Problem for Integrodifferential Systems
320(2)
11.6 The Bounded State Problem for Ordinary Differential Systems
322(4)
11.7 Further Discussion of the Bounded State Problem
326(3)
11.8 Sufficiency Conditions
329(3)
11.9 Nonlinear Beam Problem
332(5)
12 Hamilton-Jacobi Theory
337(34)
12.1 Introduction
337(1)
12.2 Problem Formulation and Assumptions
338(2)
12.3 Continuity of the Value Function
340(4)
12.4 The Lower Dini Derivate Necessary Condition
344(5)
12.5 The Value as Viscosity Solution
349(4)
12.6 Uniqueness
353(6)
12.7 The Value Function as Verification Function
359(1)
12.8 Optimal Synthesis
360(6)
12.9 The Maximum Principle
366(5)
Bibliography 371(8)
Index 379
Leonard David Berkovitz, Negash G. Medhin
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