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**Impulsive control theory.**
*(English)*
Zbl 0996.93003

Lecture Notes in Control and Information Sciences. 272. Berlin: Springer. xx, 348 p. EUR 87.50 (net); sFr 145.00; £60.00; $ 95.80 (2001).

This is the first of two graduate textbooks that are dedicated to impulsive systems and control. In this book the emphasis is put on the theoretical aspects of impulsive control systems. The existence and stability of impulsive control strategies are studied in a very detailed manner. The book is organized in a highly self-contained and reader-friendly way. Many important theorems are accompanied by detailed proofs, which in many cases are constructive. The book is divided into 11 chapters.

In Chapter 1 (Preliminaries) definitions of different kinds of impulsive control systems are presented. Also some basic knowledge on the existence and continuation of solutions to impulsive differential equations are introduced briefly. Some extensively used definitions and mathematical results are summarized in this chapter. In Chapter 2 (Linear impulsive control) the author studies the stability and the controllability of time-invariant and time-varying linear impulsive control systems. In Chapter 3 (Comparison methods) the stability of impulsive control systems is studied based on comparison methods. In Chapter 4 (Impulsive control with fixed-time impulses) different methods for designing impulsive controllers with fixed-time impulses are presented. Stability of impulsive control systems, stability of sets, and stability in terms of two measures are studied. In Chapter 5 (Impulsive control with impulses at variable time) impulsive control systems with impulses at variable time are studied by using linear decomposition methods and methods based on two measures. In Chapter 6 (Practical stability of impulsive control) the practical stability impulsive control systems are studied by using methods based on comparison and multi-comparison systems and two measures. Applications to impulsive control of nonautonomous chaotic systems are also presented. In Chapter 7 (Other impulsive control strategies) the partial stability of impulsive control systems with impulses at fixed time and variable time is studied. Also the stability of integro-differential impulsive control systems is studied, based on comparison methods, methods in terms of two measures and the practical stability concept. In Chapter 8 (Impulsive computational verb control) the basic principles of verb impulsive control are presented as a natural extension of fuzzy control. In Chapter 9 (Impulsive control of periodic motion) the author studies impulsive control of periodic motion in linear periodic autonomous and nonautonomous systems. In Chapter 10 (Impulsive control of almost periodic motions) two kinds of plants are studied, almost periodic plants and periodic plants driven by almost periodic control signals. The results can be used to control chaotic systems and design nanoelectronic circuits. In Chapter 11 (Applications to nanoelectronics) applications of impulsive control theory to nanoelectronics are presented.

In Chapter 1 (Preliminaries) definitions of different kinds of impulsive control systems are presented. Also some basic knowledge on the existence and continuation of solutions to impulsive differential equations are introduced briefly. Some extensively used definitions and mathematical results are summarized in this chapter. In Chapter 2 (Linear impulsive control) the author studies the stability and the controllability of time-invariant and time-varying linear impulsive control systems. In Chapter 3 (Comparison methods) the stability of impulsive control systems is studied based on comparison methods. In Chapter 4 (Impulsive control with fixed-time impulses) different methods for designing impulsive controllers with fixed-time impulses are presented. Stability of impulsive control systems, stability of sets, and stability in terms of two measures are studied. In Chapter 5 (Impulsive control with impulses at variable time) impulsive control systems with impulses at variable time are studied by using linear decomposition methods and methods based on two measures. In Chapter 6 (Practical stability of impulsive control) the practical stability impulsive control systems are studied by using methods based on comparison and multi-comparison systems and two measures. Applications to impulsive control of nonautonomous chaotic systems are also presented. In Chapter 7 (Other impulsive control strategies) the partial stability of impulsive control systems with impulses at fixed time and variable time is studied. Also the stability of integro-differential impulsive control systems is studied, based on comparison methods, methods in terms of two measures and the practical stability concept. In Chapter 8 (Impulsive computational verb control) the basic principles of verb impulsive control are presented as a natural extension of fuzzy control. In Chapter 9 (Impulsive control of periodic motion) the author studies impulsive control of periodic motion in linear periodic autonomous and nonautonomous systems. In Chapter 10 (Impulsive control of almost periodic motions) two kinds of plants are studied, almost periodic plants and periodic plants driven by almost periodic control signals. The results can be used to control chaotic systems and design nanoelectronic circuits. In Chapter 11 (Applications to nanoelectronics) applications of impulsive control theory to nanoelectronics are presented.

Reviewer: S.K.Ntouyas (Ioannina)

### MSC:

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

93D15 | Stabilization of systems by feedback |

93C57 | Sampled-data control/observation systems |

93B51 | Design techniques (robust design, computer-aided design, etc.) |

34A37 | Ordinary differential equations with impulses |

45J05 | Integro-ordinary differential equations |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

93C42 | Fuzzy control/observation systems |

34C25 | Periodic solutions to ordinary differential equations |