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**Stability of motion. Translated by Arne P. Baartz.**
*(English)*
Zbl 0189.38503

Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen. Band 138. Berlin-Heidelberg-New York: Springer-Verlag. xii, 446 p. with 63 figures (1967).

This book gives the reader a complete introduction to stability theory and familiarize him with the newer results of the theory and their applications to practical questions. The author treats stability theory as a mathematical discipline, characterizes its methods and proves its theorems rigorously and completely as mathematical theorems. There are many examples to illustrate the arguments and to stress the interactions between theory and practice.

Chapter I gives the stability concept in mechanics and the definitions of stability in the sense of Lyapunov. Chapter II contains the general discussion on linear differential equations with constant coefficients, linear differential difference equations and linear difference equations, and also gives criteria for stability of those equations.

In Chapter III, the fundamental concepts in autonomous systems are given and stability of a critical point in 2-dimensional systems is discussed. In Chapter IV and Chapter V, the author discusses in great detail Lyapunov’s direct method. These chapters contain all principal theorems in the author’s own formulation, construction of a Lyapunov function for a linear equation, the problem of Aizerman, absolute stability for control systems (including Popov’s method). Furthermore, the author discusses some of more recent results and applications to differential difference equations, to functional differential equations and to partial differential equations.

Chapter VI gives the converse of the stability theorems. Chapter VII gives modified stability criteria as Krasovskii’s criterion and LaSalle’s criterion. It contains also some discussion of perturbed systems by Lyapunov function for the unperturbed system (including total stability, integral stability). Chapter VIII contains Lyapunov’s reducibility theorem, the order numbers of a linear differential equation and stability in the first approximation. In Chapter IX, the Lyapunov expansion theorem is discussed, and in Chapter X, critical cases for differential equations are discussed.

Chapter XI concerns periodic and almost periodic motions. The author starts by discussing the existence and the stability of a periodic (an almost periodic) solution of a linear system and continuous his discussion to piecewise linear equations. Then periodic solutions of perturbed linear systems are discussed. This chapter contains also a brief discussion of orbital stability of a periodic solution to an autonomous differential equation.

Chapter I gives the stability concept in mechanics and the definitions of stability in the sense of Lyapunov. Chapter II contains the general discussion on linear differential equations with constant coefficients, linear differential difference equations and linear difference equations, and also gives criteria for stability of those equations.

In Chapter III, the fundamental concepts in autonomous systems are given and stability of a critical point in 2-dimensional systems is discussed. In Chapter IV and Chapter V, the author discusses in great detail Lyapunov’s direct method. These chapters contain all principal theorems in the author’s own formulation, construction of a Lyapunov function for a linear equation, the problem of Aizerman, absolute stability for control systems (including Popov’s method). Furthermore, the author discusses some of more recent results and applications to differential difference equations, to functional differential equations and to partial differential equations.

Chapter VI gives the converse of the stability theorems. Chapter VII gives modified stability criteria as Krasovskii’s criterion and LaSalle’s criterion. It contains also some discussion of perturbed systems by Lyapunov function for the unperturbed system (including total stability, integral stability). Chapter VIII contains Lyapunov’s reducibility theorem, the order numbers of a linear differential equation and stability in the first approximation. In Chapter IX, the Lyapunov expansion theorem is discussed, and in Chapter X, critical cases for differential equations are discussed.

Chapter XI concerns periodic and almost periodic motions. The author starts by discussing the existence and the stability of a periodic (an almost periodic) solution of a linear system and continuous his discussion to piecewise linear equations. Then periodic solutions of perturbed linear systems are discussed. This chapter contains also a brief discussion of orbital stability of a periodic solution to an autonomous differential equation.

Reviewer: Taro Yoshizawa (Sendai)

### MSC:

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

34Dxx | Stability theory for ordinary differential equations |

74H55 | Stability of dynamical problems in solid mechanics |

34K20 | Stability theory of functional-differential equations |

39A30 | Stability theory for difference equations |