Asymptotically-stable solution

A solution of a differential system that is stable according to Lyapunov (cf. Lyapunov stability) and that attracts all the other solutions with sufficiently close initial values. Thus, the solution

$$ x ( \tau , \xi _ {0} ),\ x( \alpha , \xi _ {0} ) = \xi _ {0} , $$

of the system

$$ \tag{* } \frac{dx}{d \tau } = f ( \tau , x ) $$

with a right-hand side $ f( \tau , \xi ) $, given for all $ \tau \geq \alpha $, $ \xi \in \mathbf R ^ {n} $, and which is such that solutions of (*) exist and are unique, will be an asymptotically-stable solution if, together with all its sufficiently close solutions

$$ x ( \tau , \xi ),\ | \xi - \xi _ {0} | \langle h ,\ h \rangle 0, $$

it is defined for all $ \tau \geq \alpha $ and if for an arbitrary $ \epsilon > 0 $ there exists a $ \delta $, $ 0 < \delta < h $, such that $ | \xi - \xi _ {0} | < \delta $ implies

$$ \| x ( \tau , \xi ) -x ( \tau , \xi _ {0} ) \| < \epsilon $$

for all $ \tau \geq \alpha $ and

$$ \| x ( \tau , \xi ) -x ( \tau , \xi _ {0} ) \| \rightarrow 0 $$

as $ \tau \rightarrow + \infty $.

The concept of an asymptotically-stable solution was introduced by A.M. Lyapunov [1]; it, together with various special types of uniform asymptotic stability, is extensively used in the theory of stability [2], [3], [4].

References

[1]	A.M. Lyapunov, "Problème général de la stabilité du mouvement" , Ann. of Math. Studies , 17 , Princeton Univ. Press (1947)
[2]	N.N. Krasovskii, "Stability of motion. Applications of Lyapunov's second method to differential systems and equations with delay" , Stanford Univ. Press (1963) (Translated from Russian)
[3]	W. Hahn, "Theorie und Anwendung der direkten Methode von Ljapunov" , Springer (1959)
[4]	N. Rouche, P. Habets, M. Laloy, "Stability theory by Liapunov's direct method" , Springer (1977)

References

[a1]	W. Hahn, "Stability of motion" , Springer (1967) pp. 422