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**Optimal control theory with economic applications.**
*(English)*
Zbl 0613.49001

Advanced Textbooks in Economics, Vol. 24. Amsterdam etc.: North-Holland. XVI, 445 p. $ 49.50; Dfl. 150.00 (1987).

This textbook gives an introduction to optimal control theory. The exposition is limited to deterministic models with dynamics described by ordinary differential equations. The organization of the book is the one mostly used for textbooks of this kind: starting with calculus of variations in Ch. 1, the authors proceed to a basic optimal control problem (Chapter 2) where necessary, sufficient and existence conditions are stated. A maximum principle for an autonomous case is proved. Some extensions (jumps in the state variables, sensitivity and infinite horizons) are considered in Chapter 3. The remaining chapters are devoted to an analysis of optimal control problems with mixed constraints as well as pure state constraints.

The book has been written with a view to the needs arising in economic applications of optimal control theory. Although some useful topics have been left out (for instance, numerical methods), the book supplies the reader with a fairly comprehensive collection of results and methods that will enable him/her to deal with a quite wide range of optimal control problems.

The authors emphasize the large number of examples where theory is applied to economic problems. To this reviewer, 75 examples are not an overwhelming amount in a book of almost 450 pages. For another approach, see G. Feichtinger and R. F. Hart [Optimale Kontrolle ökonomischer Prozesse (Berlin 1986)]. However, most examples are well- chosen and stated and the solutions are thoroughly worked out.

An important - and desirable - feature which sets this book apart from the majority of introductory textbooks in optimal control theory is the - sometimes almost pedantic - emphasis on precision in the theoretical developments.

On the whole, the book is a valuable addition to the literature on optimal control of economic processes.

The book has been written with a view to the needs arising in economic applications of optimal control theory. Although some useful topics have been left out (for instance, numerical methods), the book supplies the reader with a fairly comprehensive collection of results and methods that will enable him/her to deal with a quite wide range of optimal control problems.

The authors emphasize the large number of examples where theory is applied to economic problems. To this reviewer, 75 examples are not an overwhelming amount in a book of almost 450 pages. For another approach, see G. Feichtinger and R. F. Hart [Optimale Kontrolle ökonomischer Prozesse (Berlin 1986)]. However, most examples are well- chosen and stated and the solutions are thoroughly worked out.

An important - and desirable - feature which sets this book apart from the majority of introductory textbooks in optimal control theory is the - sometimes almost pedantic - emphasis on precision in the theoretical developments.

On the whole, the book is a valuable addition to the literature on optimal control of economic processes.

Reviewer: St.Jørgensen

### MSC:

49-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control |

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

91B62 | Economic growth models |

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

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