Infinite horizon optimal control. Theory and applications.

*(English)*Zbl 0649.49001The main problem of the research in the monograph is a qualitative analysis of the behavior of optimal trajectories, in the first place a convergence to the point of phase space which is a solution of a static optimization problem. The statements on this convergence are usually called turnpike theorems.

Chapter 1 gives examples of optimal control problems on unbounded time intervals in the field of economics, ecology and technology. The definitions of optimal control are introduced with a meaning for divergence of integrals in the optimality functional.

Chapter 2 presents necessary and sufficient conditions for optimality in the form of a maximum principle for trajectories with infinite horizon duration.

Chapter 3 presents some problems where turnpike behavior of optimal trajectories can be fixed with the help of simple devices.

Chapter 4 considers convex autonomous problems with the help of large variations of trajectories. Autonomous systems with a nonautonomous functional of optimality, where the nonautonomous component is a decreasing exponent, are considered in chapter 5.

Chapter 6 gives statements on the convergence of optimal trajectories for nonautonomous and nonconvex problems.

Chapter 7 is devoted to the existence of solutions on infinite time intervals for nonautonomous control systems.

Chapter 8 analyses optimal processes with infinite time for linear equations with distributed parameters interpreted as usual differential equations in Hilbert space.

The book will promote the development of a new trend in optimal control connected with qualitative research of optimal processes of large and infinite duration [see also, the reviewer, Avtom. Telemekh. 1983, No.9, 58-66 (1983; Zbl 0562.93055); ibid. 1981, No.8, 119-130 (1981; Zbl 0489.93036); and the reviewer together with A. I. Panasyuk, Prikl. Mat. Mekh. 49, 524-535 (1985; Zbl 0614.49015)].

Chapter 1 gives examples of optimal control problems on unbounded time intervals in the field of economics, ecology and technology. The definitions of optimal control are introduced with a meaning for divergence of integrals in the optimality functional.

Chapter 2 presents necessary and sufficient conditions for optimality in the form of a maximum principle for trajectories with infinite horizon duration.

Chapter 3 presents some problems where turnpike behavior of optimal trajectories can be fixed with the help of simple devices.

Chapter 4 considers convex autonomous problems with the help of large variations of trajectories. Autonomous systems with a nonautonomous functional of optimality, where the nonautonomous component is a decreasing exponent, are considered in chapter 5.

Chapter 6 gives statements on the convergence of optimal trajectories for nonautonomous and nonconvex problems.

Chapter 7 is devoted to the existence of solutions on infinite time intervals for nonautonomous control systems.

Chapter 8 analyses optimal processes with infinite time for linear equations with distributed parameters interpreted as usual differential equations in Hilbert space.

The book will promote the development of a new trend in optimal control connected with qualitative research of optimal processes of large and infinite duration [see also, the reviewer, Avtom. Telemekh. 1983, No.9, 58-66 (1983; Zbl 0562.93055); ibid. 1981, No.8, 119-130 (1981; Zbl 0489.93036); and the reviewer together with A. I. Panasyuk, Prikl. Mat. Mekh. 49, 524-535 (1985; Zbl 0614.49015)].

Reviewer: V.Panasyuk

##### MSC:

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

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

49K15 | Optimality conditions for problems involving ordinary differential equations |

93C15 | Control/observation systems governed by ordinary differential equations |

93C20 | Control/observation systems governed by partial differential equations |

93C25 | Control/observation systems in abstract spaces |

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

49J27 | Existence theories for problems in abstract spaces |

49K20 | Optimality conditions for problems involving partial differential equations |

49K27 | Optimality conditions for problems in abstract spaces |