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**Stability analysis of nonlinear systems.**
*(English)*
Zbl 0676.34003

The book consists of 5 chapters.

1. Inequalities. - The Gronwall-type of inequalities and their generalizations to integral and integro-differential inequalities, including inequalities for piecewise continuous functions and reaction- diffusion inequalities, are presented.

2. Variation of parameters and monotone technique. - Some formulas, including Alekseev’s one, given the relation between solutions of a differential system and of its perturbation, are presented. With help of them and of the inequalities in the first chapter, the authors succeed in estimation of the solutions. Then the method of lower and upper solutions combined with monotone iterative technique is used for proving existence results. Also stability properties are studied with help of variational systems.

3. Stability of motion in terms of two measures. - The stability in terms of two measures means that perturbation of initial values and that of corresponding solutions are measured in different scales. The method of Lyapunov functions for investigating such a stability, is discussed.

4. Stability of perturbed motion. - Methods developped in the first three chapters, are applied to investigate different problems in the stability theory.

5. Models of real world phenomena. - Examples illustrating how methods of the book can be applied for studying equations describing some real world phenomena, are given. Population models and those from analytical mechanics, economics, viscoelasticity and chemical kinetics, are considered.

In whole the book presents a good exposition of some tendencies in modern theory of the stability of motion, important contribution to which has been made by the authors.

1. Inequalities. - The Gronwall-type of inequalities and their generalizations to integral and integro-differential inequalities, including inequalities for piecewise continuous functions and reaction- diffusion inequalities, are presented.

2. Variation of parameters and monotone technique. - Some formulas, including Alekseev’s one, given the relation between solutions of a differential system and of its perturbation, are presented. With help of them and of the inequalities in the first chapter, the authors succeed in estimation of the solutions. Then the method of lower and upper solutions combined with monotone iterative technique is used for proving existence results. Also stability properties are studied with help of variational systems.

3. Stability of motion in terms of two measures. - The stability in terms of two measures means that perturbation of initial values and that of corresponding solutions are measured in different scales. The method of Lyapunov functions for investigating such a stability, is discussed.

4. Stability of perturbed motion. - Methods developped in the first three chapters, are applied to investigate different problems in the stability theory.

5. Models of real world phenomena. - Examples illustrating how methods of the book can be applied for studying equations describing some real world phenomena, are given. Population models and those from analytical mechanics, economics, viscoelasticity and chemical kinetics, are considered.

In whole the book presents a good exposition of some tendencies in modern theory of the stability of motion, important contribution to which has been made by the authors.

Reviewer: Yu.N.Bibikov

### MSC:

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

34D15 | Singular perturbations of ordinary differential equations |

34D20 | Stability of solutions to ordinary differential equations |

34A34 | Nonlinear ordinary differential equations and systems |

34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |