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.
Author(s): Leonard David Berkovitz, Negash G. Medhin
Series: Chapman & Hall/CRC Applied Mathematics and Nonlinear Science
Publisher: Chapman and Hall/CRC
Year: 2013
Language: English
Pages: xii+380
Tags: Автоматизация;Теория автоматического управления (ТАУ);
Nonlinear Optimal Control Theory......Page 4
Contents......Page 6
Foreword......Page 10
Preface......Page 12
1.2 A Problem of Production Planning......Page 14
1.3 Chemical Engineering......Page 16
1.4 Flight Mechanics......Page 17
1.5 Electrical Engineering......Page 20
1.6 The Brachistochrone Problem......Page 22
1.7 An Optimal Harvesting Problem......Page 25
1.8 Vibration of a Nonlinear Beam......Page 26
2.2 Formulation of Problems Governed by Ordinary Differential Equations......Page 28
2.3 Mathematical Formulation......Page 31
2.4 Equivalent Formulations......Page 35
2.5 Isoperimetric Problems and Parameter Optimization......Page 39
2.6 Relationship with the Calculus of Variations......Page 40
2.7 Hereditary Problems......Page 45
3.1 Introduction......Page 48
3.2 The Relaxed Problem; Compact Constraints......Page 51
3.3 Weak Compactness of Relaxed Controls......Page 56
3.4 Filippov’s Lemma......Page 69
3.5 The Relaxed Problem; Non-Compact Constraints......Page 76
3.6 The Chattering Lemma; Approximation to Relaxed Controls......Page 79
4.1 Introduction......Page 92
4.2 Non-Existence and Non-Uniqueness of Optimal Controls......Page 93
4.3 Existence of Relaxed Optimal Controls......Page 96
4.4 Existence of Ordinary Optimal Controls......Page 105
4.5 Classes of Ordinary Problems Having Solutions......Page 111
4.6 Inertial Controllers......Page 114
4.7 Systems Linear in the State Variable......Page 116
5.1 Introduction......Page 126
5.2 Properties of Set Valued Maps......Page 127
5.3 Facts from Analysis......Page 130
5.4 Existence via the Cesari Property......Page 135
5.5 Existence Without the Cesari Property......Page 152
5.6 Compact Constraints Revisited......Page 158
6.1 Introduction......Page 162
6.2 A Dynamic Programming Derivation of the Maximum Principle......Page 163
6.3 Statement of Maximum Principle......Page 172
6.4 An Example......Page 186
6.5 Relationship with the Calculus of Variations......Page 190
6.6 Systems Linear in the State Variable......Page 195
6.7 Linear Systems......Page 199
6.8 The Linear Time Optimal Problem......Page 205
6.9 Linear Plant-Quadratic Criterion Problem......Page 206
7.1 Introduction......Page 218
7.2 Penalty Proof of Necessary Conditions in Finite Dimensions......Page 220
7.3 The Norm of a Relaxed Control; Compact Constraints......Page 223
7.4 Necessary Conditions for an Unconstrained Problem......Page 225
7.5 The ε-Problem......Page 231
7.6 The ε-Maximum Principle......Page 236
7.7 The Maximum Principle; Compact Constraints......Page 241
7.8 Proof of Theorem 6.3.9......Page 247
7.9 Proof of Theorem 6.3.12......Page 251
7.10 Proof of Theorem 6.3.17 and Corollary 6.3.19......Page 253
7.11 Proof of Theorem 6.3.22......Page 257
8.2 The Rocket Car......Page 262
8.3 A Non-Linear Quadratic Example......Page 268
8.4 A Linear Problem with Non-Convex Constraints......Page 270
8.5 A Relaxed Problem......Page 272
8.6 The Brachistochrone Problem......Page 275
8.7 Flight Mechanics......Page 280
8.8 An Optimal Harvesting Problem......Page 286
8.9 Rotating Antenna Example......Page 289
9.2 Problem Statement......Page 296
9.3 Systems Linear in the State Variable......Page 298
9.5 Systems Governed by Integrodifferential Systems......Page 300
9.6 Linear Plant Quadratic Cost Criterion......Page 301
9.7 A Minimum Principle......Page 302
10.2 Problem Statement and Assumptions......Page 308
10.3 Minimum Principle......Page 309
10.4 Some Linear Systems......Page 311
10.6 Infinite Dimensional Setting......Page 313
10.6.1 Approximate Optimality Conditions......Page 315
10.6.2 Optimality Conditions......Page 317
11.2 Statement of the Problem......Page 318
11.3 є-Optimality Conditions......Page 319
11.4 Limiting Operations......Page 329
11.5 The Bounded State Problem for Integrodifferential Systems......Page 333
11.6 The Bounded State Problem for Ordinary Differential Systems......Page 335
11.7 Further Discussion of the Bounded State Problem......Page 339
11.8 Sufficiency Conditions......Page 342
11.9 Nonlinear Beam Problem......Page 345
12.1 Introduction......Page 350
12.2 Problem Formulation and Assumptions......Page 351
12.3 Continuity of the Value Function......Page 353
12.4 The Lower Dini Derivate Necessary Condition......Page 357
12.5 The Value as Viscosity Solution......Page 362
12.6 Uniqueness......Page 366
12.7 The Value Function as Verification Function......Page 372
12.8 Optimal Synthesis......Page 373
12.9 The Maximum Principle......Page 379
Bibliography......Page 384
Index......Page 392
