Turnpike properties in the calculus of variations and optimal control.

*(English)*Zbl 1100.49003
Nonconvex Optimization and its Applications 80. New York, NY: Springer (ISBN 0-387-28155-X/hbk; 0-387-28154-1/ebook). xxii, 395 p. (2006).

This monograph is dedicated to the study of the turnpike theory and is based mainly on the author’s results on the subject obtained in the last twenty years. As mentioned in the preface, “the term was first coined by Samuelson who showed that an efficient expanding economy would for most of the time be in the vicinity of a balanced equilibrium path (also called a von Neumann path)”. The book is organized in twelve chapters: Ch. 1, Infinite horizon variational problems; Ch. 2, Extremals of nonautonomous problems; Ch. 3, Extremals of autonomous problems; Ch. 4, Infinite horizon autonomous problems; Ch. 5, Turnpike for autonomous problems; Ch. 6, Linear periodic control systems, Ch. 7, Linear systems with nonperiodic integrands; Ch. 8, Discrete-time control systems; Ch. 9, Control problems in Hilbert spaces; Ch. 10, A class of differential inclusions; Ch. 11, Convex processes; Ch. 12, A dynamic zero-sum game.

The chapters have a similar structure: one begins with the statements of the main results followed by the needed auxiliary results with their proofs; then one provides the proofs of the main results. Some chapters end with examples. In the first chapter one discusses three notions of optimality for infinite horizon problems: \((f)\)-minimal solutions, overtaking optimal solutions and good solutions. The second chapter contains the most general and strongest results of the book and they refer to turnpike properties for nonautonomous nonconvex variational problems. In the third chapter one establishes a generic existence of a weakly optimal function. Moreover, it is shown that the turnpike property holds for approximate solutions on finite intervals when the integrand has the asymptotic turnpike property. In the forth chapter one improves some turnpike results from the preceding chapter; so, it is shown that all optimal solutions with initial points belonging to a bounded set converge uniformly to the turnpike. The aim of the fifth chapter is to improve the turnpike results from the second chapter in the case of autonomous variational problems on finite intervals under additional assumptions on the integrands: their smoothness and growth conditions on their partial derivatives. Chapters 6 and 7 are dedicated to the study of turnpike properties of a class of linear control problems arising in engineering. In the first of them the integrand is assumed to be periodic in the time variable and in both chapters the integrand is strictly convex in the state and control variables. In every chapter were considered discrete-time versions of the envisages control problems. In Chapter 8 one studies an autonomous problem with convex cost function on a bounded closed convex subset of a Banach space and a nonautonomous nonconvex problem on a complete metric space. In the ninth chapter one obtains the convergence to the turnpike for the weak and strong topologies, for the last one assuming that the integrand is strictly convex. The last three chapters are dedicated to the study of the turnpike property for a class of differential inclusions arising in mathematical economics, the study of the dynamic properties of optimal trajectories of convex processes, and applications of the turnpike property to game theory, respectively.

The list of references has 112 items, 23 being papers of the author. It also contains a preface, an introduction and an index. The book addresses to mathematicians working in optimal control, calculus of variations, mathematical economics and game theory.

The chapters have a similar structure: one begins with the statements of the main results followed by the needed auxiliary results with their proofs; then one provides the proofs of the main results. Some chapters end with examples. In the first chapter one discusses three notions of optimality for infinite horizon problems: \((f)\)-minimal solutions, overtaking optimal solutions and good solutions. The second chapter contains the most general and strongest results of the book and they refer to turnpike properties for nonautonomous nonconvex variational problems. In the third chapter one establishes a generic existence of a weakly optimal function. Moreover, it is shown that the turnpike property holds for approximate solutions on finite intervals when the integrand has the asymptotic turnpike property. In the forth chapter one improves some turnpike results from the preceding chapter; so, it is shown that all optimal solutions with initial points belonging to a bounded set converge uniformly to the turnpike. The aim of the fifth chapter is to improve the turnpike results from the second chapter in the case of autonomous variational problems on finite intervals under additional assumptions on the integrands: their smoothness and growth conditions on their partial derivatives. Chapters 6 and 7 are dedicated to the study of turnpike properties of a class of linear control problems arising in engineering. In the first of them the integrand is assumed to be periodic in the time variable and in both chapters the integrand is strictly convex in the state and control variables. In every chapter were considered discrete-time versions of the envisages control problems. In Chapter 8 one studies an autonomous problem with convex cost function on a bounded closed convex subset of a Banach space and a nonautonomous nonconvex problem on a complete metric space. In the ninth chapter one obtains the convergence to the turnpike for the weak and strong topologies, for the last one assuming that the integrand is strictly convex. The last three chapters are dedicated to the study of the turnpike property for a class of differential inclusions arising in mathematical economics, the study of the dynamic properties of optimal trajectories of convex processes, and applications of the turnpike property to game theory, respectively.

The list of references has 112 items, 23 being papers of the author. It also contains a preface, an introduction and an index. The book addresses to mathematicians working in optimal control, calculus of variations, mathematical economics and game theory.

Reviewer: Constantin Zălinescu (Iaşi)

##### MSC:

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

49J05 | Existence theories for free problems in one independent variable |

49J10 | Existence theories for free problems in two or more independent variables |

49J15 | Existence theories for optimal control problems involving ordinary differential equations |

49J20 | Existence theories for optimal control problems involving partial differential equations |

49J24 | Optimal control problems with differential inclusions (existence) (MSC2000) |

49J99 | Existence theories in calculus of variations and optimal control |

91A05 | 2-person games |