##
**Extensions and relaxations.**
*(English)*
Zbl 1031.93001

Mathematics and its Applications (Dordrecht). 542. Dordrecht: Kluwer Academic Publishers. xiv, 408 p. (2002).

The monograph continues the series of publications of the authors in which they study a general approach to relaxation of extremal problems. The class of problems under consideration includes problems with integral constraints (impulse constraints). The idea of the method goes back to the concept of a control strategy in positional differential games considered by N. N. Krasovskij and A. I. Subbotin [see Positional differential games (Russian). Moskau: Nauka (1974; Zbl 0298.90067)] and a related iterative construction proposed by the first author of the monograph under review [Sov. Math., Dokl. 19, 559-562 (1978); translation from Dokl. Akad. Nauk SSSR 240, 36-39 (1978; Zbl 0415.90093)]. The construction requires an extension of the concept of control strategy, which is implemented by using finitely additive measures as generalized controls.

In the first chapter, an example of a control problem with integral constraints is considered. This example is used in the further chapters as a test problem. In the following four chapters the theoretical background is developed. It includes foundations of topology, measure theory, Darboux integrals and compactification. Periodically, the presentation of the material includes examples of applications of the abstract concepts in control theory, mostly using the test example from the first chapter. In the sixth chapter, the method of programmed iterations for constructing a positional strategy is presented in a generalized form. In chapter seven, this method is further extended to multi-valued quasi-strategies. A generalized quasi-strategy acts as a multi-valued mapping defined on a space of finitely-additive measures.

The conclusion section at the end of each chapter provides a summary. In the final conclusions, the authors summarize the whole book and compare it with other publications.

In the first chapter, an example of a control problem with integral constraints is considered. This example is used in the further chapters as a test problem. In the following four chapters the theoretical background is developed. It includes foundations of topology, measure theory, Darboux integrals and compactification. Periodically, the presentation of the material includes examples of applications of the abstract concepts in control theory, mostly using the test example from the first chapter. In the sixth chapter, the method of programmed iterations for constructing a positional strategy is presented in a generalized form. In chapter seven, this method is further extended to multi-valued quasi-strategies. A generalized quasi-strategy acts as a multi-valued mapping defined on a space of finitely-additive measures.

The conclusion section at the end of each chapter provides a summary. In the final conclusions, the authors summarize the whole book and compare it with other publications.

Reviewer: D.Silin (Berkeley)

### MSC:

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

49M20 | Numerical methods of relaxation type |

28-02 | Research exposition (monographs, survey articles) pertaining to measure and integration |

28B20 | Set-valued set functions and measures; integration of set-valued functions; measurable selections |

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

93B52 | Feedback control |

91A23 | Differential games (aspects of game theory) |

49J45 | Methods involving semicontinuity and convergence; relaxation |