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**Elements of finite-dimensional systems and control theory.**
*(English)*
Zbl 0658.93002

Pitman Monographs and Surveys in Pure and Applied Mathematics, 37. Harlow (UK): Longman Scientific & Technical; New York: John Wiley & Sons. xiii, 421 p. £58.00 (1988).

The last two decades of systems theory evolution reveals that the developments in this area appear to proceed along several distinct lines and that the mathematical treatment may widely differ from an approach to another. Under these circumstances and faced with a mountain of literature, this book is a successful author’s attempt to acquaint the young researcher rapidly and rigorously with the basics of modern systems theory and design.

In this respect, after an introductory chapter which presents the most common models used to represent physical systems, chapters 2 and 3 deal with the basic properties of linear and nonlinear systems addressing the questions of existence, uniqueness (based on the Banach and Schauder fixed point theorems) and the solution dependence on initial data, on parameters, and on controls. These are the cardinal notions (remembering the qualitative theory of differential equations) which allow a unified and natural self-contained approach of the material in the following chapters.

In chapter 4 is studied the question of stability based on the Lyapunov methods including the extensions of LaSalle and Lefschetz. Chapter 5 considers the question of controllability, observability and stabilizability for linear and nonlinear systems. The subject of chapter 6 is optimal control, where LaSalle’s bang-bang principle, Pontrjagin’s principle, Bellman’s principle of optimality including some existence theorems for optimal controls are presented. Finally, chapter 7 presents the basic theory of stochastic systems and includes some prominent results on stability, filtering, control and reliability.

In view of the major outlook, most of the theoretic results are accompanied by proofs. Many illustrative and good examples and exercises drawn from the physical sciences, engineering, management sciences or the biological sciences are also given. The style is sober and precise. The intended audience of the book are the undergraduate and graduate students in mathematics, control/systems engineering, physical sciences, economics and management sciences, and it may be a valuable stepping stone towards further study and research in systems theory of applied mathematicians, control/systems engineers and systems analysts.

In this respect, after an introductory chapter which presents the most common models used to represent physical systems, chapters 2 and 3 deal with the basic properties of linear and nonlinear systems addressing the questions of existence, uniqueness (based on the Banach and Schauder fixed point theorems) and the solution dependence on initial data, on parameters, and on controls. These are the cardinal notions (remembering the qualitative theory of differential equations) which allow a unified and natural self-contained approach of the material in the following chapters.

In chapter 4 is studied the question of stability based on the Lyapunov methods including the extensions of LaSalle and Lefschetz. Chapter 5 considers the question of controllability, observability and stabilizability for linear and nonlinear systems. The subject of chapter 6 is optimal control, where LaSalle’s bang-bang principle, Pontrjagin’s principle, Bellman’s principle of optimality including some existence theorems for optimal controls are presented. Finally, chapter 7 presents the basic theory of stochastic systems and includes some prominent results on stability, filtering, control and reliability.

In view of the major outlook, most of the theoretic results are accompanied by proofs. Many illustrative and good examples and exercises drawn from the physical sciences, engineering, management sciences or the biological sciences are also given. The style is sober and precise. The intended audience of the book are the undergraduate and graduate students in mathematics, control/systems engineering, physical sciences, economics and management sciences, and it may be a valuable stepping stone towards further study and research in systems theory of applied mathematicians, control/systems engineers and systems analysts.

Reviewer: M.Voicu

### MSC:

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

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

93C05 | Linear systems in control theory |

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

49L20 | Dynamic programming in optimal control and differential games |

49K30 | Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) |

93C10 | Nonlinear systems in control theory |

93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |

93E03 | Stochastic systems in control theory (general) |

93E20 | Optimal stochastic control |