This work describes all basic equaitons and inequalities that form the necessary and sufficient optimality conditions of variational calculus and the theory of optimal control. Subjects addressed include developments in the investigation of optimality conditions, new classes of solutions, analytical and computation methods, and applications.
Author(s): Vadim Krotov
Series: Monographs and textbooks in pure and applied mathematics 195
Edition: 1
Publisher: M. Dekker
Year: 1996
Language: English
Pages: 399
City: New York
Tags: Математика;Методы оптимизации;
Title......Page 1
Publication Data ......Page 2
Preface ......Page 3
Contents ......Page 11
1.1 Preliminary Description of the Problem ......Page 16
1.2 Controlled Processes ......Page 18
1.3 Optimal Control of Ordinary Process. Problem Statement and Some Applications ......Page 20
1.4 Optimal Control of Multistage Processes ......Page 30
1.5 Optimal Control of Multiargument Processes. Problem Statement and Some Applications ......Page 31
References ......Page 33
2.1 Some Preliminary Theses. Extension Principle ......Page 36
2.2 Principle of Decomposition with Respect to Time ......Page 38
2.3 Equivalent Representations ......Page 40
2.4 Direct Application of Equivalent Representations of Some Optimal Control Problems ......Page 43
2.5 Global Bounds ......Page 50
2.6 Sufficient Optimality Conditions ......Page 58
2.7 Sufficient Optimality Conditions. Nonfixed Boundary State ......Page 62
2.8 Control Synthesis ......Page 64
2.9 Estimations of Achievability Regions. Strengthened Lower Bounds and Sufficient Optimality Conditions ......Page 65
2.10 Principle of Optimality ......Page 67
Commentary and Bibliography ......Page 69
References ......Page 73
3. SOME SPECIAL OPTIMAL CONTROL PROBLEMS.DIRECT APPLICATION OF THE SUFFICIENT OPTIMALITY CONDITIONS ......Page 76
3.1 Some Generalizations of the Degenerate Euler's Problem ......Page 77
3.2 Simplest Macromodel of Optimal Economic Development ......Page 86
3.3 Two-Dimensional Autonomous Problem Linear with Respect to Control ......Page 96
3.4 Optimal Control for Vertical Maneuvers of Aircraft ......Page 105
3.5 Multidimensional Degenerate Problem ......Page 126
3.6 Applied Physics Problem: Maximization of Atomic Excitement by Resonance Field ......Page 132
Commentary and Bibliography ......Page 136
References ......Page 143
4.1 Some Useful Formulas ......Page 148
4.2 Sufficient Optimality Conditions and Maximum Principle ......Page 149
4.3 General Boundary Conditions ......Page 153
4.4 Absolute Minimum Problem in Lagrange's Method ......Page 154
4.5 Optimal Control for Linear Systems ......Page 156
4.6 Strong Local Minimum ......Page 161
4.7 Local Minimum Conditions for Euler's Problem of Variational Calculus ......Page 164
4.8 Principle of Least Action ......Page 169
4.9 Second-Order Conditions for Degenerate Problems ......Page 172
4.10 Processes with Discrete Argument ......Page 175
4.11 Multiargument Systems ......Page 183
Commentary and Bibliography ......Page 192
References ......Page 193
5. OPTIMAL FEEDBACK POLICY. HAMILTON-JACOBI METHOD ......Page 196
5.1 Equations for Optimal Control Synthesis ......Page 199
5.2 Algorithmic Features of the Method ......Page 203
5.3 Synthesis of Controllers for Linear Systems with a Quadratic Criterion ......Page 204
5.4 Approximate Optimal Synthesis ......Page 209
5.5 Descent of Spacecraft into Atmosphere Using the Engine: Approximate Optimal Control Synthesis ......Page 216
5.6 Approximate Optimal Controllers for Linear Systems with Restricted Control ......Page 222
5.7 Analytical Synthesis of Optimal Control for Nonlinear Systems ......Page 234
References ......Page 249
6. COMPUTER METHODS FOR SUCCESSIVE IMPROVEMENTS IN A CONTROL PROGRAM ......Page 252
6.1 Infinitesimal Methods of Control Improvement ......Page 254
6.2 Global Methods of Successive Improvements of Control ......Page 257
6.3 Control of Observation Process ......Page 264
6.4 Some Special Problems ......Page 267
6.5 Optimal Control of Quantum Systems by Laser Radiation ......Page 268
6.6 Knapsack Multivariate Problem ......Page 274
6.7 Comparative Discussion of the Iterative Methods ......Page 276
References ......Page 277
7.1 Problem Statement. Basic Idea of the Method ......Page 280
7.2 Method of Improvement ......Page 284
7.3 Methods of Realization of an Elementary Operation ......Page 288
7.4 Convergence ......Page 292
7.5 Examples ......Page 296
Appendix: Some Auxiliary Results ......Page 302
References ......Page 306
8. EXTENSION OF THE CLASS OF SOLVING FUNCTIONS. THEOREMS OF EXISTENCE OF SOLVING FUNCTIONS ......Page 308
8.1 Global Necessary and Sufficient Conditions of Optimality for Multistage Processes ......Page 310
8.2 Necessary and Sufficient Optimality Conditions for Ordinary Processes Free of State Restrictions ......Page 313
8.3 Weak Solving Functions ......Page 318
8.4 Global Necessary and Sufficient Conditions of Optimality for Ordinary Controlled Processes with State Constraints ......Page 321
8.5 Global Necessary and Sufficient Conditions of Optimality for Multiargument Controlled Processes ......Page 327
8.6 Some Concluding Observations ......Page 332
References ......Page 334
9.1 Problem Statement ......Page 338
9.2 Global Bounds and Conditions of Optimality ......Page 343
9.3 Differential Games with Complete Information on the State. Optimal Synthesis ......Page 353
9.4 Differential Games with Complete Information on the State. Equations of Optimal Trajectory ......Page 359
9.5 Incomplete Information on State. Conditions for Lack of Information Price ......Page 365
9.6 Information Differential Games. Discrete Sufficient Information Operator ......Page 368
9.7 Information Differential Games. Optimal Control for Insufficient Information ......Page 373
9.8 Invariance Problem. Conditions of Invariance ......Page 381
9.9 Invariance of a Linearly Quadratic Functional in a Quasilinear System ......Page 384
9.10 Absolute Invariance ......Page 385
9.11 Synthesis of Invariant Systems ......Page 386
9.12 Invariant Control of Spacecraft Descent in a Planetary Atmosphere ......Page 387
Commentary and Bibliography ......Page 391
References ......Page 393
Index ......Page 396
