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**Time-delay systems. Analysis, optimization and applications.**
*(English)*
Zbl 0658.93001

North-Holland Systems and Control Series, Vol. 9. Amsterdam etc.: North- Holland (Elsevier Science Publishers B.V.) XVI, 504 p.; $ 89.00; Dfl. 200.00 (1987).

This book is a good introduction to the qualitative theory of dynamic systems with after-effect. It starts with several mathematical models of actual time-delay processes. The first part deals with the mathematical description of time-delay systems. Large-scale time-delay systems and a linearization of nonlinear time-delay systems as well as state space and frequency domain representations are extensively considered here.

The second part gives an analysis of linear time-delay systems. It considers various solution properties, the fundamental matrix, adjoint state equations and other properties. Special attention is given to the problem of stability and controllability of time-delay systems. Such important problems as uniform asymptotic stability, the Lyapunov method, stability of stochastic time-delay systems, stability of time-delay systems with variable delays, Pontryagin’s stability method, several controllability and observability criteria are described in this part. Unfortunately, some interesting results concerning criteria of multipoint controllability, modal controllability (control) by using both integral and difference linear regulators, rank conditions for stabilization and others are not presented here.

Part III: Optimization (Chapter 6. Optimization of time-delay systems. Chapter 7. Suboptimal control of time-delay systems, Chapter 8. Near- optimum design on large-scale time-delay systems) includes, in particular, the following areas: the maximum principle, a generalized Riccati method, the dynamic programming method, a sensitivity approach, a hierarchical control and a computational algorithm. The next part is concerned with various applications of time-delay systems to gold rolling mills, water resources and others.

There are two appendices containing reviews on linear algebra, on the Laplace transform and modified z-transforms at the end of the book.

The second part gives an analysis of linear time-delay systems. It considers various solution properties, the fundamental matrix, adjoint state equations and other properties. Special attention is given to the problem of stability and controllability of time-delay systems. Such important problems as uniform asymptotic stability, the Lyapunov method, stability of stochastic time-delay systems, stability of time-delay systems with variable delays, Pontryagin’s stability method, several controllability and observability criteria are described in this part. Unfortunately, some interesting results concerning criteria of multipoint controllability, modal controllability (control) by using both integral and difference linear regulators, rank conditions for stabilization and others are not presented here.

Part III: Optimization (Chapter 6. Optimization of time-delay systems. Chapter 7. Suboptimal control of time-delay systems, Chapter 8. Near- optimum design on large-scale time-delay systems) includes, in particular, the following areas: the maximum principle, a generalized Riccati method, the dynamic programming method, a sensitivity approach, a hierarchical control and a computational algorithm. The next part is concerned with various applications of time-delay systems to gold rolling mills, water resources and others.

There are two appendices containing reviews on linear algebra, on the Laplace transform and modified z-transforms at the end of the book.

Reviewer: V.Marcenko

### MSC:

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

34K35 | Control problems for functional-differential equations |

93C30 | Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) |

34K99 | Functional-differential equations (including equations with delayed, advanced or state-dependent argument) |

34K20 | Stability theory of functional-differential equations |

49K99 | Optimality conditions |

93B03 | Attainable sets, reachability |

93B05 | Controllability |

93B07 | Observability |

93B35 | Sensitivity (robustness) |

93B40 | Computational methods in systems theory (MSC2010) |

93C05 | Linear systems in control theory |

93C10 | Nonlinear systems in control theory |

93A15 | Large-scale systems |

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

93D20 | Asymptotic stability in control theory |

93A13 | Hierarchical systems |

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

93E15 | Stochastic stability in control theory |

93E20 | Optimal stochastic control |

93C95 | Application models in control theory |