This book is devoted to the study of classes of optimal control problems arising in economic growth theory, related to the Robinson-Solow-Srinivasan (RSS) model. The model was introduced in the 1960s by economists Joan Robinson, Robert Solow, and Thirukodikaval Nilakanta Srinivasan and was further studied by Robinson, Nobuo Okishio, and Joseph Stiglitz. Since then, the study of the RSS model has become an important element of economic dynamics. In this book, two large general classes of optimal control problems, both of them containing the RSS model as a particular case, are presented for study. For these two classes, a turnpike theory is developed and the existence of solutions to the corresponding infinite horizon optimal control problems is established.
The book contains 9 chapters. Chapter 1 discusses turnpike properties for some optimal control problems that are known in the literature, including problems corresponding to the RSS model. The first class of optimal control problems is studied in Chaps. 2-6. In Chap. 2, infinite horizon optimal control problems with nonautonomous optimality criteria are considered. The utility functions, which determine the optimality criterion, are nonconcave. This class of models contains the RSS model as a particular case. The stability of the turnpike phenomenon of the one-dimensional nonautonomous concave RSS model is analyzed in Chap. 3. The following chapter takes up the study of a class of autonomous nonconcave optimal control problems, a subclass of problems considered in Chap. 2. The equivalence of the turnpike property and the asymptotic turnpike property, as well as the stability of the turnpike phenomenon, is established. Turnpike conditions and the stability of the turnpike phenomenon for nonautonomous problems are examined in Chap. 5, with Chap. 6 devoted to the study of the turnpike properties for the one-dimensional nonautonomous nonconcave RSS model. The utility functions, which determine the optimality criterion, are nonconcave. The class of RSS models is identified with a complete metric space of utility functions. Using the Baire category approach, the turnpike phenomenon is shown to hold for most of the models. Chapter 7 begins the study of the second large class of autonomous optimal control problems, and turnpike conditions are established. The stability of the turnpike phenomenon for this class of problems is investigated further in Chaps. 8 and 9.
Chapter 1. Introduction
1.1 The turnpike phenomenon
1.2. Nonconcave (nonconvex) problems
1.3. Examples
1.4. Stability of the turnpike phenomenon
1.5 The Robinson-Solow-Srinivasan model
1.6. Overtaking optimal programs for the Robinson-Solow-Srinivasan model
1.7. Turnpike properties of the Robinson-Solow-Srinivasan model
Chapter 2. Good programs for infinite horizon optimal control problems
2.1. Preliminaries and the main results
2.2. Upper semicontinuity of cost functions
2.3 The nonstationary Robinson-Solow-Srinivasan model
2.4. Auxiliary results for Theorems 2.4, 2.5 and 2.7
2.5. Properties of the function $U$
2.6. Proof of Theorem 2.4
2.7. Proof of Theorem 2.5
2.8. Proof of Theorem 2.7
2.9. Problems with discounting
2.10. The Robinson-Solow-Srinivasan model with discounting
2.11. Auxiliary results for Theorem 2.23
2.12. Proof of Theorem 2.23
Chapter 3. One-dimensional concave Robinson-Solow-Srinivasan model
3.1. Preliminaries and main results
3.2. Auxiliary results
3.3. Proof of Theorem 3.14
3.4. Stability results
3.5. Proof of Theorem 3.26
Chapter 4. Autonomous nonconcave optimal control problems
4.1. Preliminaries
4.2. A controlability lemma
4.3. The turnpike property implies the asymptotic turnpike property
4.4. Auxiliary results
4.5. The asymptotic turnpike property implies the turnpike property
4.6. A weak turnpike property
4.7. A turnpike result
4.8. Auxiliary results for Theorem 4.11
4.9. Proof of Theorem 4.11
4.10. Stability results
4.11. A subclass of models
Chapter 5. Turnpike phenomenon for nonautonomous problems
5.1 Preliminaries
5.2. Turnpike properties
5.3. Examples
5.4. The turnpike property implies (P1) and (P2)
5.5. Auxiliary results
5.6. Completion of the proof of Theorem 5.2
5.7. A turnpike result
5.8. An auxiliary result for Theorem 5.8
5.9. Proof of Theorem 5.8
5.10. Stability results
5.11. Proof of Theorem 5.11
Chapter 6. Generic results
6.1. One-dimensional nonconcave Robinson-Solow-Srinivasan model
6.2. The main results
6.3. Auxiliary results
6.4. Auxiliary results for Theorem 6.6.
6.5. Proof of Theorem 6.6
6.6. Proof of Theorem 6.10
Chapter 7. Turnpike phenomenon for a class of optimal control problems
7.1. Preliminaries and main results
7.2. Auxiliary results
7.3. Proof of Theorem 7.2
7.4. Proof of Theorem 7.3
7.5. Proofs of Theorems 7.4 and 7.5
7.7. The Robinson-Solow-Srinivasan model
7.8. A general model of economic dynamics
7.9. Equivalence of optimality criterions
7.10. Proof of Theorem 7.22
7.11 Weak turnpike results
7.12. Proof of Theorem 7.23
Chapter 8. Problems with perturbed objective functions
8.1 Preliminaries and main results
8.2. Auxiliary results
8.3 Proofs of Theorems 8.1 and 8.2
8.4. Proofs of Theorems 8.3 and 8.4
8.5. Problems with discounting
8.6. Auxiliary results for Theorems 8.10 and 8.11
8.7. Proof of Theorem 8.10
8.8. Proof of Theorem 8.11
8.9. Existence of overtaking optimal programs
Chapter 9. Stability of the turnpike phenomenon
9.1. Preliminaries and main results
9.2. Examples
9.3. Auxiliary results
9.4. Proof of Theorem 9.3
9.5. Proof of Theorem 9.4
9.6. Problems with discounting
9.7. Auxiliary results for Theorem 9.15
9.8. Proof of Theorem 9.15
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