Retarded dynamical systems: stability and characteristic functions.

*(English)*Zbl 0686.34044
Pitman Research Notes in Mathematics Series, 210. Harlow: Longman Scientific & Technical; New York: John Wiley & Sons, Inc. 151 p. £16.50 (1989).

The book contains an overview of work of the author in the last ten years and presents stability properties of dynamical systems with memory. The first chapter is designed to give the reader a brief survey of the basic stability criteria for linear autonomous functional differential equations (FDEs). The second chapter is devoted to necessary and sufficient conditions for stability of retarded FDE’s whose analysis is based on characteristic functions. Some of these results are generalized to a class of neutral systems. The chapter ends with a short discussion on the stability of the trivial solutions of nonlinear FDEs. Chapter 3 contains the stability charts of linear FDEs useful to investigate practical problems related to systems with slight damping terms. The last chapter contains a sample of the applications of the theory here presented. The author first studies a predator-prey system of population dynamics of retarded type, useful to understand the latter analysis which deals with stability problems related to practical examples arising from biomechanics, robotics and machine dynamics. The numerous examples and remarks of the book make its presentation clear and amply demonstrate the significance of this stability theory for the applications.

Reviewer: P.Pucci

##### MSC:

37-XX | Dynamical systems and ergodic theory |

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

34K20 | Stability theory of functional-differential equations |

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

92D25 | Population dynamics (general) |

93D15 | Stabilization of systems by feedback |