##
**Dynamic systems and control with applications.**
*(English)*
Zbl 1127.93001

Hackensack, NJ: World Scientific (ISBN 981-270-053-6/hbk). xv, 450 p. (2006).

This is intended to be a textbook for graduate students of Engineering, Mathematics, Management, Finance. However, it is more than a textbook, since it includes not only classical material, but also elements of modern analysis as well as new topics, such as impulsive systems.

The main goal is a comprehensive treatment of system theory and optimal control. For the sake of completness some basic prerequisites are also included. This is obvious from the list of chapters: 1. Basic mathematical background; 2. Linear systems; 3. Nonlinear systems; 4. Basic stability theory; 5. Observability and identification; 6. Controllability and stabilizability; 7. Basic calculus of variations; 8. Optimal control: necessary conditions and existence; 9. Linear quadratic regulator theory; 10. Time optimal control; 11. Stochastic systems with applications; A. Basic results from analysis; Bibliography; Index. Some applications are included within the text to illustrate the theoretical results. These applications are discussed from a mathematical point of view. However, some remarks on the models involved are also added to remind the reader of practical aspects that motivate the whole theory.

The material covers the main topics in the field, including some prerequisites. So the book is sufficiently complete. It is written in a nice style, resulting from a combination of mathematical equations and explanatory remarks. However, it should be pointed out that the mathematical construction is sometimes weak or even incorrect. So the book is good for readers who already know the basics of the theory. To illustrate my observation, let me give just one example: Theorem 5.5.2 (page 167) is not true. Indeed, the rank condition \((5.5.56)\) is not sufficient for identifiability (detectability), as proved by my PhD student Tihomir Gyulov, who constructed the following simple counter-example: \(A\) is the matrix with rows \((-1,0,0)\), \((0,0,0)\), \((0,0,1)\); \(B\) is the matrix with rows \((1,0)\), \((0,1)\), \((0,0)\); \(H=(1,1,0)\); and \(u(t)= \text{col} (t, -1+e^{-t})\). Verification of this assertion is left to the reader.

The main goal is a comprehensive treatment of system theory and optimal control. For the sake of completness some basic prerequisites are also included. This is obvious from the list of chapters: 1. Basic mathematical background; 2. Linear systems; 3. Nonlinear systems; 4. Basic stability theory; 5. Observability and identification; 6. Controllability and stabilizability; 7. Basic calculus of variations; 8. Optimal control: necessary conditions and existence; 9. Linear quadratic regulator theory; 10. Time optimal control; 11. Stochastic systems with applications; A. Basic results from analysis; Bibliography; Index. Some applications are included within the text to illustrate the theoretical results. These applications are discussed from a mathematical point of view. However, some remarks on the models involved are also added to remind the reader of practical aspects that motivate the whole theory.

The material covers the main topics in the field, including some prerequisites. So the book is sufficiently complete. It is written in a nice style, resulting from a combination of mathematical equations and explanatory remarks. However, it should be pointed out that the mathematical construction is sometimes weak or even incorrect. So the book is good for readers who already know the basics of the theory. To illustrate my observation, let me give just one example: Theorem 5.5.2 (page 167) is not true. Indeed, the rank condition \((5.5.56)\) is not sufficient for identifiability (detectability), as proved by my PhD student Tihomir Gyulov, who constructed the following simple counter-example: \(A\) is the matrix with rows \((-1,0,0)\), \((0,0,0)\), \((0,0,1)\); \(B\) is the matrix with rows \((1,0)\), \((0,1)\), \((0,0)\); \(H=(1,1,0)\); and \(u(t)= \text{col} (t, -1+e^{-t})\). Verification of this assertion is left to the reader.

Reviewer: Gheorghe Moroşanu (Budapest)

### MSC:

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

34-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations |

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

34A37 | Ordinary differential equations with impulses |

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

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

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