Stability Theory by Liapunov’s Direct Method pp 201-240 | Cite as

# A Survey of Qualitative Concepts

Chapter

## Abstract

Up to this point, we have studied a small number of concepts such as stability, attractivity, asymptotic stability, etc. They all pertain to the origin of ℛ with a continuous second member defined on I × Ω with values in ℛ

^{n}for a differential equation$$ \mathop{x}\limits^{.} = f\left( {t,x} \right) $$

(1.1)

^{n}(the symbols I and O have the same meaning here as in Section IV.2). An exception was the orbital stability, as described in Section 1.1.4: this is properly the stability of a set. It appears that the attractivity or asymptotic stability of a set are natural concepts fitting many practical applications: several examples will be seen in Chapter VII. But as soon as a set, and especially an unbounded set, is substituted to a point, namely the origin, the number of potentially useful concepts increases rapidly: in fact several types of ‘uniformities’ with respect to spatial coordinates have to be considered in addition to the familiar uniformity with respect to t_{0}(the one which makes the distinction between stability and uniform stability). The number of concepts increases to such extent that one may well wonder which ones should be studied, and which ones schouldn’t.## Keywords

Asymptotic Stability Global Asymptotic Stability Equivalence Theorem Uniform Stability Global Attractivity
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag, New York Inc. 1977