This book is devoted to the recent progress on the turnpike theory. The turnpike property was discovered by Paul A. Samuelson, who applied it to problems in mathematical economics in 1949. These properties were studied for optimal trajectories of models of economic dynamics determined by convex processes. In this monograph the author, a leading expert in modern turnpike theory, presents a number of results concerning the turnpike properties in the calculus of variations and optimal control which were obtained in the last ten years. These results show that the turnpike properties form a general phenomenon which holds for various classes of variational problems and optimal control problems. The book should help to correct the misapprehension that turnpike properties are only special features of some narrow classes of convex problems of mathematical economics.
Author(s): Alexander J. Zaslavski
Series: Nonconvex Optimization and Its Applications
Edition: 1
Publisher: Springer
Year: 2005
Language: English
Pages: 407
City: Berlin
Turnpike.Properties.in.the.Calculus.of.Variations.and.Optima. Control.Volumn.80......Page 1
Contents......Page 6
Preface......Page 10
Introduction......Page 13
1.1. Preliminaries......Page 23
1.2. Main results......Page 25
1.3. Auxiliary results......Page 29
1.4. Discrete-time control systems......Page 39
1.5. Proofs of Theorems 1.1-1.3......Page 42
2.1. Main results......Page 54
2.2. Preliminary lemmas......Page 58
2.3. Proofs of Theorems 2.1.1-2.1.4......Page 75
2.4. Periodic variational problems......Page 80
2.5. Spaces of smooth integrands......Page 83
2.6. Examples......Page 90
3.1. Main results......Page 91
3.2. Proof of Proposition 3.1.1......Page 96
3.3. Weakened version of Theorem 3.1.3......Page 99
3.4. Continuity of the function Uf (T1, T2, x, y)......Page 103
3.5. Discrete-time control systems......Page 108
3.6. Proof of Theorem 3.1.2......Page 110
3.7. Preliminary lemmas for Theorem 3.1.1......Page 114
3.8. Preliminary lemmas for Theorems 3.1.3 and 3.1.4......Page 119
3.9. Proof of Theorem 3.1.4......Page 126
3.10. Proof of Theorem 3.1.3......Page 132
3.12. Examples......Page 134
4.1. Main results......Page 135
4.2. Proofs of Theorems 4.1.1-4.1.3......Page 139
4.3. Proof of Theorem 4.1.4......Page 170
5.1. Main results......Page 172
5.2. Proof of Theorem 5.1.1......Page 177
5.3. Proof of Theorem 5.1.2......Page 188
5.4. Examples......Page 191
6.1. Main results......Page 192
6.2. Preliminary results......Page 195
6.3. Discrete-time control systems......Page 202
6.4. Proof of Theorem 6.1.1......Page 205
6.5. Proof of Theorem 6.1.2......Page 207
6.6. Proof of Theorem 6.1.3......Page 209
6.7. Proof of Theorem 6.1.4......Page 212
7.1. Main results......Page 215
7.2. Preliminary results......Page 219
7.3. Discrete-time control systems......Page 221
7.4. Proof of Theorem 7.1.1......Page 222
7.5. Proof of Theorem 7.1.2......Page 227
7.6. Proofs of Theorems 7.1.3 and 7.1.4......Page 233
