Some methods in the mathematical analysis of systems and their control.

*(English)*Zbl 0542.93034
Beijing, China: Science Press; New York: Gordon and Breach, Science Publishers, Inc. XXIII, 542 p. $ 93.50 (1981).

This book is concerned with analysis and control of the systems governed by partial differential equations (such systems are called distributed parameter systems).

The book consists of two parts. The first part is devoted to decide whether or not the problem is ”well-posed”, the problem is ”tractible” (i.e. the problem can be approximated numerically), the problem has some simpler approximation. The author presents asymptotic methods aimed at giving approximations for the solutions of various problems arising in mathematical physics, such as composite materials, perforated materials, porous media, systems with time-scale of different magnitude, etc.

The second part is devoted to the control of the system. The problem consists in finding the control function (the optimal control) which achieves in the best possible way a given objective (a cost function). The author also treats a variant of this problem obtained when there are several control functions which can be chosen either with conflicting interests or with some priority between them (game theory, Stackelberg’s strategies, etc.). Moreover, he gives an introduction to the theory of optimal control of non well posed distributed parameter systems.

He indicates several open problems at the end of each chapter of the book.

The 7 chapters are as follows: Chapter 1. Asymptotic methods in periodic structures. Chapter 2. Some problems connected with Navier-Stokes equations. Chapter 3. Some remarks on the reduction of complexity in the analysis of systems. Chapter 4. Optimal control of distributed systems. Chapter 5. Reduction of complexity in the optimal control of distributed systems. Chapter 6. Introduction to some aspects of game theory for distributed systems. Chapter 7. Optimal control of non-well-posed systems.

The book consists of two parts. The first part is devoted to decide whether or not the problem is ”well-posed”, the problem is ”tractible” (i.e. the problem can be approximated numerically), the problem has some simpler approximation. The author presents asymptotic methods aimed at giving approximations for the solutions of various problems arising in mathematical physics, such as composite materials, perforated materials, porous media, systems with time-scale of different magnitude, etc.

The second part is devoted to the control of the system. The problem consists in finding the control function (the optimal control) which achieves in the best possible way a given objective (a cost function). The author also treats a variant of this problem obtained when there are several control functions which can be chosen either with conflicting interests or with some priority between them (game theory, Stackelberg’s strategies, etc.). Moreover, he gives an introduction to the theory of optimal control of non well posed distributed parameter systems.

He indicates several open problems at the end of each chapter of the book.

The 7 chapters are as follows: Chapter 1. Asymptotic methods in periodic structures. Chapter 2. Some problems connected with Navier-Stokes equations. Chapter 3. Some remarks on the reduction of complexity in the analysis of systems. Chapter 4. Optimal control of distributed systems. Chapter 5. Reduction of complexity in the optimal control of distributed systems. Chapter 6. Introduction to some aspects of game theory for distributed systems. Chapter 7. Optimal control of non-well-posed systems.

Reviewer: T.Kobuyashi

##### MSC:

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

35B37 | PDE in connection with control problems (MSC2000) |

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

93-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to systems and control theory |

35A35 | Theoretical approximation in context of PDEs |

35C20 | Asymptotic expansions of solutions to PDEs |

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

65M99 | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |

65N99 | Numerical methods for partial differential equations, boundary value problems |

91A99 | Game theory |